\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Preface}}{23}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }The \textsf {GAP} System}}{23}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Authors and Maintainers}}{25}{section.1.2}
\contentsline {section}{\numberline {1.3}\leavevmode {\color {Chapter }Acknowledgements}}{26}{section.1.3}
\contentsline {section}{\numberline {1.4}\leavevmode {\color {Chapter }Copyright and License}}{26}{section.1.4}
\contentsline {section}{\numberline {1.5}\leavevmode {\color {Chapter }Further Information about \textsf {GAP}}}{27}{section.1.5}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }The Help System}}{28}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }Invoking the Help}}{28}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Browsing through the Sections}}{28}{section.2.2}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }Changing the Help Viewer}}{29}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}\leavevmode {\color {Chapter }SetHelpViewer}}{30}{subsection.2.3.1}
\contentsline {section}{\numberline {2.4}\leavevmode {\color {Chapter }The Pager Command}}{31}{section.2.4}
\contentsline {subsection}{\numberline {2.4.1}\leavevmode {\color {Chapter }Pager}}{31}{subsection.2.4.1}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Running GAP}}{33}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Command Line Options}}{33}{section.3.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }The gap.ini and gaprc files}}{37}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color {Chapter }The gap.ini file}}{38}{subsection.3.2.1}
\contentsline {subsection}{\numberline {3.2.2}\leavevmode {\color {Chapter }The gaprc file}}{38}{subsection.3.2.2}
\contentsline {subsection}{\numberline {3.2.3}\leavevmode {\color {Chapter }Configuring User preferences}}{39}{subsection.3.2.3}
\contentsline {subsection}{\numberline {3.2.4}\leavevmode {\color {Chapter }DeclareUserPreference}}{40}{subsection.3.2.4}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }Saving and Loading a Workspace}}{41}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter }SaveWorkspace}}{41}{subsection.3.3.1}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter }Testing for the System Architecture}}{42}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color {Chapter }ARCH{\textunderscore }IS{\textunderscore }UNIX}}{42}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}\leavevmode {\color {Chapter }ARCH{\textunderscore }IS{\textunderscore }MAC{\textunderscore }OS{\textunderscore }X}}{42}{subsection.3.4.2}
\contentsline {subsection}{\numberline {3.4.3}\leavevmode {\color {Chapter }ARCH{\textunderscore }IS{\textunderscore }WINDOWS}}{42}{subsection.3.4.3}
\contentsline {section}{\numberline {3.5}\leavevmode {\color {Chapter }Global Values that Control the \textsf {GAP} Session}}{42}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}\leavevmode {\color {Chapter }GAPInfo}}{42}{subsection.3.5.1}
\contentsline {section}{\numberline {3.6}\leavevmode {\color {Chapter }Coloring the Prompt and Input}}{42}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}\leavevmode {\color {Chapter }ColorPrompt}}{42}{subsection.3.6.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }The Programming Language}}{44}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Language Overview}}{44}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Lexical Structure}}{45}{section.4.2}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Symbols}}{45}{section.4.3}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Whitespaces}}{46}{section.4.4}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Keywords}}{47}{section.4.5}
\contentsline {section}{\numberline {4.6}\leavevmode {\color {Chapter }Identifiers}}{47}{section.4.6}
\contentsline {subsection}{\numberline {4.6.1}\leavevmode {\color {Chapter }IsValidIdentifier}}{48}{subsection.4.6.1}
\contentsline {section}{\numberline {4.7}\leavevmode {\color {Chapter }Expressions}}{48}{section.4.7}
\contentsline {section}{\numberline {4.8}\leavevmode {\color {Chapter }Variables}}{49}{section.4.8}
\contentsline {subsection}{\numberline {4.8.1}\leavevmode {\color {Chapter }IsBound (for a global variable)}}{50}{subsection.4.8.1}
\contentsline {subsection}{\numberline {4.8.2}\leavevmode {\color {Chapter }Unbind (unbind a variable)}}{50}{subsection.4.8.2}
\contentsline {section}{\numberline {4.9}\leavevmode {\color {Chapter }More About Global Variables}}{50}{section.4.9}
\contentsline {subsection}{\numberline {4.9.1}\leavevmode {\color {Chapter }IsReadOnlyGlobal}}{51}{subsection.4.9.1}
\contentsline {subsection}{\numberline {4.9.2}\leavevmode {\color {Chapter }MakeReadOnlyGlobal}}{51}{subsection.4.9.2}
\contentsline {subsection}{\numberline {4.9.3}\leavevmode {\color {Chapter }MakeReadWriteGlobal}}{51}{subsection.4.9.3}
\contentsline {subsection}{\numberline {4.9.4}\leavevmode {\color {Chapter }MakeConstantGlobal}}{52}{subsection.4.9.4}
\contentsline {subsection}{\numberline {4.9.5}\leavevmode {\color {Chapter }ValueGlobal}}{52}{subsection.4.9.5}
\contentsline {subsection}{\numberline {4.9.6}\leavevmode {\color {Chapter }IsBoundGlobal}}{52}{subsection.4.9.6}
\contentsline {subsection}{\numberline {4.9.7}\leavevmode {\color {Chapter }UnbindGlobal}}{52}{subsection.4.9.7}
\contentsline {subsection}{\numberline {4.9.8}\leavevmode {\color {Chapter }BindGlobal}}{52}{subsection.4.9.8}
\contentsline {subsection}{\numberline {4.9.9}\leavevmode {\color {Chapter }NamesGVars}}{53}{subsection.4.9.9}
\contentsline {subsection}{\numberline {4.9.10}\leavevmode {\color {Chapter }NamesSystemGVars}}{53}{subsection.4.9.10}
\contentsline {subsection}{\numberline {4.9.11}\leavevmode {\color {Chapter }NamesUserGVars}}{53}{subsection.4.9.11}
\contentsline {section}{\numberline {4.10}\leavevmode {\color {Chapter }Namespaces for \textsf {GAP} packages}}{54}{section.4.10}
\contentsline {section}{\numberline {4.11}\leavevmode {\color {Chapter }Function Calls}}{54}{section.4.11}
\contentsline {subsection}{\numberline {4.11.1}\leavevmode {\color {Chapter }Function Call With Arguments}}{54}{subsection.4.11.1}
\contentsline {subsection}{\numberline {4.11.2}\leavevmode {\color {Chapter }Function Call With Options}}{55}{subsection.4.11.2}
\contentsline {section}{\numberline {4.12}\leavevmode {\color {Chapter }Comparisons}}{56}{section.4.12}
\contentsline {section}{\numberline {4.13}\leavevmode {\color {Chapter }Arithmetic Operators}}{57}{section.4.13}
\contentsline {section}{\numberline {4.14}\leavevmode {\color {Chapter }Statements}}{58}{section.4.14}
\contentsline {section}{\numberline {4.15}\leavevmode {\color {Chapter }Assignments}}{59}{section.4.15}
\contentsline {section}{\numberline {4.16}\leavevmode {\color {Chapter }Procedure Calls}}{59}{section.4.16}
\contentsline {section}{\numberline {4.17}\leavevmode {\color {Chapter }If}}{60}{section.4.17}
\contentsline {section}{\numberline {4.18}\leavevmode {\color {Chapter }While}}{61}{section.4.18}
\contentsline {section}{\numberline {4.19}\leavevmode {\color {Chapter }Repeat}}{62}{section.4.19}
\contentsline {section}{\numberline {4.20}\leavevmode {\color {Chapter }For}}{62}{section.4.20}
\contentsline {section}{\numberline {4.21}\leavevmode {\color {Chapter }Break}}{64}{section.4.21}
\contentsline {section}{\numberline {4.22}\leavevmode {\color {Chapter }Continue}}{64}{section.4.22}
\contentsline {section}{\numberline {4.23}\leavevmode {\color {Chapter }Function}}{65}{section.4.23}
\contentsline {section}{\numberline {4.24}\leavevmode {\color {Chapter }Return (With or without Value)}}{69}{section.4.24}
\contentsline {section}{\numberline {4.25}\leavevmode {\color {Chapter }Syntax Trees}}{69}{section.4.25}
\contentsline {subsection}{\numberline {4.25.1}\leavevmode {\color {Chapter }SyntaxTree}}{69}{subsection.4.25.1}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Functions}}{70}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }Information about a function}}{70}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }NameFunction}}{70}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }NumberArgumentsFunction}}{71}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }NamesLocalVariablesFunction}}{71}{subsection.5.1.3}
\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }FilenameFunc}}{71}{subsection.5.1.4}
\contentsline {subsection}{\numberline {5.1.5}\leavevmode {\color {Chapter }StartlineFunc}}{72}{subsection.5.1.5}
\contentsline {subsection}{\numberline {5.1.6}\leavevmode {\color {Chapter }LocationFunc}}{72}{subsection.5.1.6}
\contentsline {subsection}{\numberline {5.1.7}\leavevmode {\color {Chapter }PageSource}}{72}{subsection.5.1.7}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Calling a function with a list argument that is interpreted as several arguments}}{73}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }CallFuncList}}{73}{subsection.5.2.1}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Wrapping a function, so the values produced are cached}}{74}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color {Chapter }MemoizePosIntFunction}}{74}{subsection.5.3.1}
\contentsline {section}{\numberline {5.4}\leavevmode {\color {Chapter }Functions that do nothing}}{75}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}\leavevmode {\color {Chapter }ReturnTrue}}{75}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}\leavevmode {\color {Chapter }ReturnFalse}}{75}{subsection.5.4.2}
\contentsline {subsection}{\numberline {5.4.3}\leavevmode {\color {Chapter }ReturnFail}}{76}{subsection.5.4.3}
\contentsline {subsection}{\numberline {5.4.4}\leavevmode {\color {Chapter }ReturnNothing}}{76}{subsection.5.4.4}
\contentsline {subsection}{\numberline {5.4.5}\leavevmode {\color {Chapter }ReturnFirst}}{76}{subsection.5.4.5}
\contentsline {subsection}{\numberline {5.4.6}\leavevmode {\color {Chapter }IdFunc}}{76}{subsection.5.4.6}
\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Function Types}}{77}{section.5.5}
\contentsline {subsection}{\numberline {5.5.1}\leavevmode {\color {Chapter }IsFunction}}{77}{subsection.5.5.1}
\contentsline {subsection}{\numberline {5.5.2}\leavevmode {\color {Chapter }FunctionsFamily}}{77}{subsection.5.5.2}
\contentsline {section}{\numberline {5.6}\leavevmode {\color {Chapter }Naming Conventions}}{77}{section.5.6}
\contentsline {section}{\numberline {5.7}\leavevmode {\color {Chapter }Code annotations (pragmas)}}{78}{section.5.7}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Main Loop and Break Loop}}{80}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Main Loop}}{80}{section.6.1}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Special Rules for Input Lines}}{82}{section.6.2}
\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }View and Print}}{82}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}\leavevmode {\color {Chapter }Default delegations in the library}}{83}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}\leavevmode {\color {Chapter }Recommendations for the implementation}}{83}{subsection.6.3.2}
\contentsline {subsection}{\numberline {6.3.3}\leavevmode {\color {Chapter }View}}{84}{subsection.6.3.3}
\contentsline {subsection}{\numberline {6.3.4}\leavevmode {\color {Chapter }Print}}{84}{subsection.6.3.4}
\contentsline {subsection}{\numberline {6.3.5}\leavevmode {\color {Chapter }ViewObj}}{85}{subsection.6.3.5}
\contentsline {subsection}{\numberline {6.3.6}\leavevmode {\color {Chapter }Display}}{85}{subsection.6.3.6}
\contentsline {subsection}{\numberline {6.3.7}\leavevmode {\color {Chapter }SetNameObject}}{86}{subsection.6.3.7}
\contentsline {section}{\numberline {6.4}\leavevmode {\color {Chapter }Break Loops}}{86}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}\leavevmode {\color {Chapter }quit from a break loop}}{87}{subsection.6.4.1}
\contentsline {subsection}{\numberline {6.4.2}\leavevmode {\color {Chapter }return from a break loop}}{87}{subsection.6.4.2}
\contentsline {subsection}{\numberline {6.4.3}\leavevmode {\color {Chapter }OnBreak}}{87}{subsection.6.4.3}
\contentsline {subsection}{\numberline {6.4.4}\leavevmode {\color {Chapter }OnBreakMessage}}{89}{subsection.6.4.4}
\contentsline {subsection}{\numberline {6.4.5}\leavevmode {\color {Chapter }Where}}{90}{subsection.6.4.5}
\contentsline {section}{\numberline {6.5}\leavevmode {\color {Chapter }Variable Access in a Break Loop}}{91}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}\leavevmode {\color {Chapter }DownEnv and UpEnv}}{91}{subsection.6.5.1}
\contentsline {section}{\numberline {6.6}\leavevmode {\color {Chapter }Error and ErrorCount}}{92}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}\leavevmode {\color {Chapter }Error}}{92}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}\leavevmode {\color {Chapter }ErrorNoReturn}}{93}{subsection.6.6.2}
\contentsline {subsection}{\numberline {6.6.3}\leavevmode {\color {Chapter }ErrorCount}}{93}{subsection.6.6.3}
\contentsline {section}{\numberline {6.7}\leavevmode {\color {Chapter }Leaving GAP}}{93}{section.6.7}
\contentsline {subsection}{\numberline {6.7.1}\leavevmode {\color {Chapter }QUIT}}{93}{subsection.6.7.1}
\contentsline {subsection}{\numberline {6.7.2}\leavevmode {\color {Chapter }GAP{\textunderscore }EXIT{\textunderscore }CODE}}{93}{subsection.6.7.2}
\contentsline {subsection}{\numberline {6.7.3}\leavevmode {\color {Chapter }QUIT{\textunderscore }GAP}}{93}{subsection.6.7.3}
\contentsline {subsection}{\numberline {6.7.4}\leavevmode {\color {Chapter }FORCE{\textunderscore }QUIT{\textunderscore }GAP}}{93}{subsection.6.7.4}
\contentsline {subsection}{\numberline {6.7.5}\leavevmode {\color {Chapter }InstallAtExit}}{94}{subsection.6.7.5}
\contentsline {subsection}{\numberline {6.7.6}\leavevmode {\color {Chapter }SaveOnExitFile}}{94}{subsection.6.7.6}
\contentsline {section}{\numberline {6.8}\leavevmode {\color {Chapter }Line Editing}}{94}{section.6.8}
\contentsline {section}{\numberline {6.9}\leavevmode {\color {Chapter }Editing using the \texttt {readline} library}}{97}{section.6.9}
\contentsline {subsection}{\numberline {6.9.1}\leavevmode {\color {Chapter }Readline customization}}{97}{subsection.6.9.1}
\contentsline {subsection}{\numberline {6.9.2}\leavevmode {\color {Chapter }The command line history}}{98}{subsection.6.9.2}
\contentsline {subsection}{\numberline {6.9.3}\leavevmode {\color {Chapter }SaveCommandLineHistory}}{98}{subsection.6.9.3}
\contentsline {subsection}{\numberline {6.9.4}\leavevmode {\color {Chapter }Writing your own command line editing functions}}{98}{subsection.6.9.4}
\contentsline {section}{\numberline {6.10}\leavevmode {\color {Chapter }Editing Files}}{100}{section.6.10}
\contentsline {subsection}{\numberline {6.10.1}\leavevmode {\color {Chapter }Edit}}{100}{subsection.6.10.1}
\contentsline {section}{\numberline {6.11}\leavevmode {\color {Chapter }Editor Support}}{100}{section.6.11}
\contentsline {section}{\numberline {6.12}\leavevmode {\color {Chapter }Changing the Screen Size}}{101}{section.6.12}
\contentsline {subsection}{\numberline {6.12.1}\leavevmode {\color {Chapter }SizeScreen}}{101}{subsection.6.12.1}
\contentsline {section}{\numberline {6.13}\leavevmode {\color {Chapter }Teaching Mode}}{101}{section.6.13}
\contentsline {subsection}{\numberline {6.13.1}\leavevmode {\color {Chapter }TeachingMode}}{101}{subsection.6.13.1}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Debugging and Profiling Facilities}}{103}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Recovery from NoMethodFound-Errors}}{103}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}\leavevmode {\color {Chapter }ShowArguments}}{103}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}\leavevmode {\color {Chapter }ShowArgument}}{104}{subsection.7.1.2}
\contentsline {subsection}{\numberline {7.1.3}\leavevmode {\color {Chapter }ShowDetails}}{104}{subsection.7.1.3}
\contentsline {subsection}{\numberline {7.1.4}\leavevmode {\color {Chapter }ShowMethods}}{104}{subsection.7.1.4}
\contentsline {subsection}{\numberline {7.1.5}\leavevmode {\color {Chapter }ShowOtherMethods}}{104}{subsection.7.1.5}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Inspecting Applicable Methods}}{105}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }ApplicableMethod}}{105}{subsection.7.2.1}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }Tracing Methods}}{105}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }TraceMethods (for operations)}}{105}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }TraceAllMethods}}{106}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }UntraceMethods (for operations)}}{106}{subsection.7.3.3}
\contentsline {subsection}{\numberline {7.3.4}\leavevmode {\color {Chapter }UntraceAllMethods}}{106}{subsection.7.3.4}
\contentsline {subsection}{\numberline {7.3.5}\leavevmode {\color {Chapter }TraceImmediateMethods}}{106}{subsection.7.3.5}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }Info Functions}}{107}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }NewInfoClass}}{108}{subsection.7.4.1}
\contentsline {subsection}{\numberline {7.4.2}\leavevmode {\color {Chapter }DeclareInfoClass}}{108}{subsection.7.4.2}
\contentsline {subsection}{\numberline {7.4.3}\leavevmode {\color {Chapter }SetInfoLevel}}{108}{subsection.7.4.3}
\contentsline {subsection}{\numberline {7.4.4}\leavevmode {\color {Chapter }InfoLevel}}{108}{subsection.7.4.4}
\contentsline {subsection}{\numberline {7.4.5}\leavevmode {\color {Chapter }ShowUsedInfoClasses}}{108}{subsection.7.4.5}
\contentsline {subsection}{\numberline {7.4.6}\leavevmode {\color {Chapter }Info}}{108}{subsection.7.4.6}
\contentsline {subsection}{\numberline {7.4.7}\leavevmode {\color {Chapter }Customizing \texttt {Info} (\ref {Info}) statements}}{109}{subsection.7.4.7}
\contentsline {subsection}{\numberline {7.4.8}\leavevmode {\color {Chapter }InfoWarning}}{110}{subsection.7.4.8}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Assertions}}{110}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}\leavevmode {\color {Chapter }SetAssertionLevel}}{110}{subsection.7.5.1}
\contentsline {subsection}{\numberline {7.5.2}\leavevmode {\color {Chapter }AssertionLevel}}{110}{subsection.7.5.2}
\contentsline {subsection}{\numberline {7.5.3}\leavevmode {\color {Chapter }Assert}}{110}{subsection.7.5.3}
\contentsline {section}{\numberline {7.6}\leavevmode {\color {Chapter }Timing}}{111}{section.7.6}
\contentsline {subsection}{\numberline {7.6.1}\leavevmode {\color {Chapter }Runtimes}}{111}{subsection.7.6.1}
\contentsline {subsection}{\numberline {7.6.2}\leavevmode {\color {Chapter }Runtime}}{112}{subsection.7.6.2}
\contentsline {subsection}{\numberline {7.6.3}\leavevmode {\color {Chapter }NanosecondsSinceEpoch}}{112}{subsection.7.6.3}
\contentsline {subsection}{\numberline {7.6.4}\leavevmode {\color {Chapter }time}}{112}{subsection.7.6.4}
\contentsline {subsection}{\numberline {7.6.5}\leavevmode {\color {Chapter }Sleep}}{112}{subsection.7.6.5}
\contentsline {section}{\numberline {7.7}\leavevmode {\color {Chapter }Tracking Memory Usage}}{113}{section.7.7}
\contentsline {subsection}{\numberline {7.7.1}\leavevmode {\color {Chapter }TotalMemoryAllocated}}{113}{subsection.7.7.1}
\contentsline {subsection}{\numberline {7.7.2}\leavevmode {\color {Chapter }memory{\textunderscore }allocated}}{113}{subsection.7.7.2}
\contentsline {section}{\numberline {7.8}\leavevmode {\color {Chapter }Profiling}}{113}{section.7.8}
\contentsline {subsection}{\numberline {7.8.1}\leavevmode {\color {Chapter }Function Profiling}}{113}{subsection.7.8.1}
\contentsline {subsection}{\numberline {7.8.2}\leavevmode {\color {Chapter }ProfileGlobalFunctions}}{114}{subsection.7.8.2}
\contentsline {subsection}{\numberline {7.8.3}\leavevmode {\color {Chapter }ProfileOperations}}{114}{subsection.7.8.3}
\contentsline {subsection}{\numberline {7.8.4}\leavevmode {\color {Chapter }ProfileOperationsAndMethods}}{114}{subsection.7.8.4}
\contentsline {subsection}{\numberline {7.8.5}\leavevmode {\color {Chapter }ProfileFunctions}}{114}{subsection.7.8.5}
\contentsline {subsection}{\numberline {7.8.6}\leavevmode {\color {Chapter }UnprofileFunctions}}{114}{subsection.7.8.6}
\contentsline {subsection}{\numberline {7.8.7}\leavevmode {\color {Chapter }ProfileMethods}}{115}{subsection.7.8.7}
\contentsline {subsection}{\numberline {7.8.8}\leavevmode {\color {Chapter }UnprofileMethods}}{115}{subsection.7.8.8}
\contentsline {subsection}{\numberline {7.8.9}\leavevmode {\color {Chapter }DisplayProfile}}{115}{subsection.7.8.9}
\contentsline {subsection}{\numberline {7.8.10}\leavevmode {\color {Chapter }ClearProfile}}{115}{subsection.7.8.10}
\contentsline {subsection}{\numberline {7.8.11}\leavevmode {\color {Chapter }An Example of Function Profiling}}{116}{subsection.7.8.11}
\contentsline {subsection}{\numberline {7.8.12}\leavevmode {\color {Chapter }Line By Line Profiling}}{119}{subsection.7.8.12}
\contentsline {subsection}{\numberline {7.8.13}\leavevmode {\color {Chapter }Line by Line profiling example}}{119}{subsection.7.8.13}
\contentsline {subsection}{\numberline {7.8.14}\leavevmode {\color {Chapter }ProfileLineByLine}}{120}{subsection.7.8.14}
\contentsline {subsection}{\numberline {7.8.15}\leavevmode {\color {Chapter }CoverageLineByLine}}{120}{subsection.7.8.15}
\contentsline {subsection}{\numberline {7.8.16}\leavevmode {\color {Chapter }UnprofileLineByLine}}{120}{subsection.7.8.16}
\contentsline {subsection}{\numberline {7.8.17}\leavevmode {\color {Chapter }UncoverageLineByLine}}{120}{subsection.7.8.17}
\contentsline {subsection}{\numberline {7.8.18}\leavevmode {\color {Chapter }ActivateProfileColour}}{121}{subsection.7.8.18}
\contentsline {subsection}{\numberline {7.8.19}\leavevmode {\color {Chapter }IsLineByLineProfileActive}}{121}{subsection.7.8.19}
\contentsline {subsection}{\numberline {7.8.20}\leavevmode {\color {Chapter }DisplayCacheStats}}{121}{subsection.7.8.20}
\contentsline {subsection}{\numberline {7.8.21}\leavevmode {\color {Chapter }ClearCacheStats}}{121}{subsection.7.8.21}
\contentsline {section}{\numberline {7.9}\leavevmode {\color {Chapter }Information about the version used}}{121}{section.7.9}
\contentsline {section}{\numberline {7.10}\leavevmode {\color {Chapter }Test Files}}{121}{section.7.10}
\contentsline {subsection}{\numberline {7.10.1}\leavevmode {\color {Chapter }Starting and stopping test}}{121}{subsection.7.10.1}
\contentsline {subsection}{\numberline {7.10.2}\leavevmode {\color {Chapter }Test}}{122}{subsection.7.10.2}
\contentsline {subsection}{\numberline {7.10.3}\leavevmode {\color {Chapter }TestDirectory}}{125}{subsection.7.10.3}
\contentsline {section}{\numberline {7.11}\leavevmode {\color {Chapter }Debugging Recursion}}{126}{section.7.11}
\contentsline {subsection}{\numberline {7.11.1}\leavevmode {\color {Chapter }SetRecursionTrapInterval}}{127}{subsection.7.11.1}
\contentsline {section}{\numberline {7.12}\leavevmode {\color {Chapter }Global Memory Information}}{128}{section.7.12}
\contentsline {subsection}{\numberline {7.12.1}\leavevmode {\color {Chapter }GasmanStatistics}}{128}{subsection.7.12.1}
\contentsline {subsection}{\numberline {7.12.2}\leavevmode {\color {Chapter }GasmanMessageStatus}}{129}{subsection.7.12.2}
\contentsline {subsection}{\numberline {7.12.3}\leavevmode {\color {Chapter }GasmanLimits}}{129}{subsection.7.12.3}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }Options Stack}}{130}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }Functions Dealing with the Options Stack}}{130}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\leavevmode {\color {Chapter }PushOptions}}{130}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}\leavevmode {\color {Chapter }PopOptions}}{131}{subsection.8.1.2}
\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }ResetOptionsStack}}{131}{subsection.8.1.3}
\contentsline {subsection}{\numberline {8.1.4}\leavevmode {\color {Chapter }OnQuit}}{131}{subsection.8.1.4}
\contentsline {subsection}{\numberline {8.1.5}\leavevmode {\color {Chapter }ValueOption}}{131}{subsection.8.1.5}
\contentsline {subsection}{\numberline {8.1.6}\leavevmode {\color {Chapter }DisplayOptionsStack}}{131}{subsection.8.1.6}
\contentsline {subsection}{\numberline {8.1.7}\leavevmode {\color {Chapter }InfoOptions}}{131}{subsection.8.1.7}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Options Stack {\textendash } an Example}}{132}{section.8.2}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Files and Filenames}}{133}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter }Portability}}{133}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}\leavevmode {\color {Chapter }LastSystemError}}{133}{subsection.9.1.1}
\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }GAP Root Directories}}{133}{section.9.2}
\contentsline {section}{\numberline {9.3}\leavevmode {\color {Chapter }Directories}}{134}{section.9.3}
\contentsline {subsection}{\numberline {9.3.1}\leavevmode {\color {Chapter }IsDirectory}}{134}{subsection.9.3.1}
\contentsline {subsection}{\numberline {9.3.2}\leavevmode {\color {Chapter }Directory}}{134}{subsection.9.3.2}
\contentsline {subsection}{\numberline {9.3.3}\leavevmode {\color {Chapter }DirectoryTemporary}}{134}{subsection.9.3.3}
\contentsline {subsection}{\numberline {9.3.4}\leavevmode {\color {Chapter }DirectoryCurrent}}{135}{subsection.9.3.4}
\contentsline {subsection}{\numberline {9.3.5}\leavevmode {\color {Chapter }DirectoriesLibrary}}{135}{subsection.9.3.5}
\contentsline {subsection}{\numberline {9.3.6}\leavevmode {\color {Chapter }DirectoriesSystemPrograms}}{135}{subsection.9.3.6}
\contentsline {subsection}{\numberline {9.3.7}\leavevmode {\color {Chapter }DirectoryContents}}{135}{subsection.9.3.7}
\contentsline {subsection}{\numberline {9.3.8}\leavevmode {\color {Chapter }DirectoryDesktop}}{135}{subsection.9.3.8}
\contentsline {subsection}{\numberline {9.3.9}\leavevmode {\color {Chapter }DirectoryHome}}{136}{subsection.9.3.9}
\contentsline {section}{\numberline {9.4}\leavevmode {\color {Chapter }File Names}}{136}{section.9.4}
\contentsline {subsection}{\numberline {9.4.1}\leavevmode {\color {Chapter }Filename}}{136}{subsection.9.4.1}
\contentsline {section}{\numberline {9.5}\leavevmode {\color {Chapter }Special Filenames}}{137}{section.9.5}
\contentsline {section}{\numberline {9.6}\leavevmode {\color {Chapter }File Access}}{137}{section.9.6}
\contentsline {subsection}{\numberline {9.6.1}\leavevmode {\color {Chapter }IsExistingFile}}{137}{subsection.9.6.1}
\contentsline {subsection}{\numberline {9.6.2}\leavevmode {\color {Chapter }IsReadableFile}}{137}{subsection.9.6.2}
\contentsline {subsection}{\numberline {9.6.3}\leavevmode {\color {Chapter }IsWritableFile}}{138}{subsection.9.6.3}
\contentsline {subsection}{\numberline {9.6.4}\leavevmode {\color {Chapter }IsExecutableFile}}{138}{subsection.9.6.4}
\contentsline {subsection}{\numberline {9.6.5}\leavevmode {\color {Chapter }IsDirectoryPath}}{138}{subsection.9.6.5}
\contentsline {section}{\numberline {9.7}\leavevmode {\color {Chapter }File Operations}}{138}{section.9.7}
\contentsline {subsection}{\numberline {9.7.1}\leavevmode {\color {Chapter }Read}}{138}{subsection.9.7.1}
\contentsline {subsection}{\numberline {9.7.2}\leavevmode {\color {Chapter }ReadAsFunction}}{139}{subsection.9.7.2}
\contentsline {subsection}{\numberline {9.7.3}\leavevmode {\color {Chapter }PrintTo and AppendTo}}{139}{subsection.9.7.3}
\contentsline {subsection}{\numberline {9.7.4}\leavevmode {\color {Chapter }LogTo}}{140}{subsection.9.7.4}
\contentsline {subsection}{\numberline {9.7.5}\leavevmode {\color {Chapter }InputLogTo}}{140}{subsection.9.7.5}
\contentsline {subsection}{\numberline {9.7.6}\leavevmode {\color {Chapter }OutputLogTo}}{140}{subsection.9.7.6}
\contentsline {subsection}{\numberline {9.7.7}\leavevmode {\color {Chapter }CrcFile}}{140}{subsection.9.7.7}
\contentsline {subsection}{\numberline {9.7.8}\leavevmode {\color {Chapter }RemoveFile}}{141}{subsection.9.7.8}
\contentsline {subsection}{\numberline {9.7.9}\leavevmode {\color {Chapter }UserHomeExpand}}{141}{subsection.9.7.9}
\contentsline {subsection}{\numberline {9.7.10}\leavevmode {\color {Chapter }Reread}}{141}{subsection.9.7.10}
\contentsline {chapter}{\numberline {10}\leavevmode {\color {Chapter }Streams}}{142}{chapter.10}
\contentsline {section}{\numberline {10.1}\leavevmode {\color {Chapter }Categories for Streams and the StreamsFamily}}{142}{section.10.1}
\contentsline {subsection}{\numberline {10.1.1}\leavevmode {\color {Chapter }IsStream}}{142}{subsection.10.1.1}
\contentsline {subsection}{\numberline {10.1.2}\leavevmode {\color {Chapter }IsClosedStream}}{142}{subsection.10.1.2}
\contentsline {subsection}{\numberline {10.1.3}\leavevmode {\color {Chapter }IsInputStream}}{143}{subsection.10.1.3}
\contentsline {subsection}{\numberline {10.1.4}\leavevmode {\color {Chapter }IsInputTextStream}}{143}{subsection.10.1.4}
\contentsline {subsection}{\numberline {10.1.5}\leavevmode {\color {Chapter }IsInputTextNone}}{143}{subsection.10.1.5}
\contentsline {subsection}{\numberline {10.1.6}\leavevmode {\color {Chapter }IsOutputStream}}{143}{subsection.10.1.6}
\contentsline {subsection}{\numberline {10.1.7}\leavevmode {\color {Chapter }IsOutputTextStream}}{143}{subsection.10.1.7}
\contentsline {subsection}{\numberline {10.1.8}\leavevmode {\color {Chapter }IsOutputTextNone}}{143}{subsection.10.1.8}
\contentsline {subsection}{\numberline {10.1.9}\leavevmode {\color {Chapter }StreamsFamily}}{143}{subsection.10.1.9}
\contentsline {section}{\numberline {10.2}\leavevmode {\color {Chapter }Operations applicable to All Streams}}{144}{section.10.2}
\contentsline {subsection}{\numberline {10.2.1}\leavevmode {\color {Chapter }CloseStream}}{144}{subsection.10.2.1}
\contentsline {subsection}{\numberline {10.2.2}\leavevmode {\color {Chapter }FileDescriptorOfStream}}{144}{subsection.10.2.2}
\contentsline {subsection}{\numberline {10.2.3}\leavevmode {\color {Chapter }UNIXSelect}}{144}{subsection.10.2.3}
\contentsline {section}{\numberline {10.3}\leavevmode {\color {Chapter }Operations for Input Streams}}{144}{section.10.3}
\contentsline {subsection}{\numberline {10.3.1}\leavevmode {\color {Chapter }Read (for streams)}}{145}{subsection.10.3.1}
\contentsline {subsection}{\numberline {10.3.2}\leavevmode {\color {Chapter }ReadAsFunction (for streams)}}{145}{subsection.10.3.2}
\contentsline {subsection}{\numberline {10.3.3}\leavevmode {\color {Chapter }ReadByte}}{145}{subsection.10.3.3}
\contentsline {subsection}{\numberline {10.3.4}\leavevmode {\color {Chapter }ReadLine}}{145}{subsection.10.3.4}
\contentsline {subsection}{\numberline {10.3.5}\leavevmode {\color {Chapter }ReadAll}}{146}{subsection.10.3.5}
\contentsline {subsection}{\numberline {10.3.6}\leavevmode {\color {Chapter }IsEndOfStream}}{146}{subsection.10.3.6}
\contentsline {subsection}{\numberline {10.3.7}\leavevmode {\color {Chapter }PositionStream}}{147}{subsection.10.3.7}
\contentsline {subsection}{\numberline {10.3.8}\leavevmode {\color {Chapter }RewindStream}}{147}{subsection.10.3.8}
\contentsline {subsection}{\numberline {10.3.9}\leavevmode {\color {Chapter }SeekPositionStream}}{147}{subsection.10.3.9}
\contentsline {section}{\numberline {10.4}\leavevmode {\color {Chapter }Operations for Output Streams}}{147}{section.10.4}
\contentsline {subsection}{\numberline {10.4.1}\leavevmode {\color {Chapter }WriteByte}}{147}{subsection.10.4.1}
\contentsline {subsection}{\numberline {10.4.2}\leavevmode {\color {Chapter }WriteLine}}{147}{subsection.10.4.2}
\contentsline {subsection}{\numberline {10.4.3}\leavevmode {\color {Chapter }WriteAll}}{148}{subsection.10.4.3}
\contentsline {subsection}{\numberline {10.4.4}\leavevmode {\color {Chapter }PrintTo and AppendTo (for streams)}}{148}{subsection.10.4.4}
\contentsline {subsection}{\numberline {10.4.5}\leavevmode {\color {Chapter }LogTo (for streams)}}{148}{subsection.10.4.5}
\contentsline {subsection}{\numberline {10.4.6}\leavevmode {\color {Chapter }InputLogTo (for streams)}}{148}{subsection.10.4.6}
\contentsline {subsection}{\numberline {10.4.7}\leavevmode {\color {Chapter }OutputLogTo (for streams)}}{149}{subsection.10.4.7}
\contentsline {subsection}{\numberline {10.4.8}\leavevmode {\color {Chapter }SetPrintFormattingStatus}}{149}{subsection.10.4.8}
\contentsline {section}{\numberline {10.5}\leavevmode {\color {Chapter }File Streams}}{150}{section.10.5}
\contentsline {subsection}{\numberline {10.5.1}\leavevmode {\color {Chapter }InputTextFile}}{150}{subsection.10.5.1}
\contentsline {subsection}{\numberline {10.5.2}\leavevmode {\color {Chapter }OutputTextFile}}{150}{subsection.10.5.2}
\contentsline {section}{\numberline {10.6}\leavevmode {\color {Chapter }User Streams}}{151}{section.10.6}
\contentsline {subsection}{\numberline {10.6.1}\leavevmode {\color {Chapter }InputTextUser}}{151}{subsection.10.6.1}
\contentsline {subsection}{\numberline {10.6.2}\leavevmode {\color {Chapter }OutputTextUser}}{151}{subsection.10.6.2}
\contentsline {subsection}{\numberline {10.6.3}\leavevmode {\color {Chapter }InputFromUser}}{151}{subsection.10.6.3}
\contentsline {section}{\numberline {10.7}\leavevmode {\color {Chapter }String Streams}}{151}{section.10.7}
\contentsline {subsection}{\numberline {10.7.1}\leavevmode {\color {Chapter }InputTextString}}{152}{subsection.10.7.1}
\contentsline {subsection}{\numberline {10.7.2}\leavevmode {\color {Chapter }OutputTextString}}{152}{subsection.10.7.2}
\contentsline {section}{\numberline {10.8}\leavevmode {\color {Chapter }Input-Output Streams}}{152}{section.10.8}
\contentsline {subsection}{\numberline {10.8.1}\leavevmode {\color {Chapter }IsInputOutputStream}}{153}{subsection.10.8.1}
\contentsline {subsection}{\numberline {10.8.2}\leavevmode {\color {Chapter }InputOutputLocalProcess}}{153}{subsection.10.8.2}
\contentsline {subsection}{\numberline {10.8.3}\leavevmode {\color {Chapter }ReadAllLine}}{153}{subsection.10.8.3}
\contentsline {section}{\numberline {10.9}\leavevmode {\color {Chapter }Dummy Streams}}{154}{section.10.9}
\contentsline {subsection}{\numberline {10.9.1}\leavevmode {\color {Chapter }InputTextNone}}{154}{subsection.10.9.1}
\contentsline {subsection}{\numberline {10.9.2}\leavevmode {\color {Chapter }OutputTextNone}}{154}{subsection.10.9.2}
\contentsline {section}{\numberline {10.10}\leavevmode {\color {Chapter }Handling of Streams in the Background}}{154}{section.10.10}
\contentsline {subsection}{\numberline {10.10.1}\leavevmode {\color {Chapter }InstallCharReadHookFunc}}{155}{subsection.10.10.1}
\contentsline {subsection}{\numberline {10.10.2}\leavevmode {\color {Chapter }UnInstallCharReadHookFunc}}{155}{subsection.10.10.2}
\contentsline {section}{\numberline {10.11}\leavevmode {\color {Chapter }Comma separated files}}{155}{section.10.11}
\contentsline {subsection}{\numberline {10.11.1}\leavevmode {\color {Chapter }ReadCSV}}{155}{subsection.10.11.1}
\contentsline {subsection}{\numberline {10.11.2}\leavevmode {\color {Chapter }PrintCSV}}{156}{subsection.10.11.2}
\contentsline {chapter}{\numberline {11}\leavevmode {\color {Chapter }Processes}}{157}{chapter.11}
\contentsline {section}{\numberline {11.1}\leavevmode {\color {Chapter }Process and Exec}}{157}{section.11.1}
\contentsline {subsection}{\numberline {11.1.1}\leavevmode {\color {Chapter }Process}}{157}{subsection.11.1.1}
\contentsline {subsection}{\numberline {11.1.2}\leavevmode {\color {Chapter }Exec}}{159}{subsection.11.1.2}
\contentsline {chapter}{\numberline {12}\leavevmode {\color {Chapter }Objects and Elements}}{160}{chapter.12}
\contentsline {section}{\numberline {12.1}\leavevmode {\color {Chapter }Objects}}{160}{section.12.1}
\contentsline {subsection}{\numberline {12.1.1}\leavevmode {\color {Chapter }IsObject}}{160}{subsection.12.1.1}
\contentsline {section}{\numberline {12.2}\leavevmode {\color {Chapter }Elements as equivalence classes}}{160}{section.12.2}
\contentsline {section}{\numberline {12.3}\leavevmode {\color {Chapter }Sets}}{161}{section.12.3}
\contentsline {section}{\numberline {12.4}\leavevmode {\color {Chapter }Domains}}{161}{section.12.4}
\contentsline {section}{\numberline {12.5}\leavevmode {\color {Chapter }Identical Objects}}{161}{section.12.5}
\contentsline {subsection}{\numberline {12.5.1}\leavevmode {\color {Chapter }IsIdenticalObj}}{161}{subsection.12.5.1}
\contentsline {subsection}{\numberline {12.5.2}\leavevmode {\color {Chapter }IsNotIdenticalObj}}{162}{subsection.12.5.2}
\contentsline {section}{\numberline {12.6}\leavevmode {\color {Chapter }Mutability and Copyability}}{163}{section.12.6}
\contentsline {subsection}{\numberline {12.6.1}\leavevmode {\color {Chapter }IsCopyable}}{163}{subsection.12.6.1}
\contentsline {subsection}{\numberline {12.6.2}\leavevmode {\color {Chapter }IsMutable}}{163}{subsection.12.6.2}
\contentsline {subsection}{\numberline {12.6.3}\leavevmode {\color {Chapter }Immutable}}{164}{subsection.12.6.3}
\contentsline {subsection}{\numberline {12.6.4}\leavevmode {\color {Chapter }MakeImmutable}}{164}{subsection.12.6.4}
\contentsline {subsection}{\numberline {12.6.5}\leavevmode {\color {Chapter }Mutability of Iterators}}{164}{subsection.12.6.5}
\contentsline {subsection}{\numberline {12.6.6}\leavevmode {\color {Chapter }Mutability of Results of Arithmetic Operations}}{164}{subsection.12.6.6}
\contentsline {section}{\numberline {12.7}\leavevmode {\color {Chapter }Duplication of Objects}}{165}{section.12.7}
\contentsline {subsection}{\numberline {12.7.1}\leavevmode {\color {Chapter }ShallowCopy}}{165}{subsection.12.7.1}
\contentsline {subsection}{\numberline {12.7.2}\leavevmode {\color {Chapter }StructuralCopy}}{165}{subsection.12.7.2}
\contentsline {section}{\numberline {12.8}\leavevmode {\color {Chapter }Other Operations Applicable to any Object}}{166}{section.12.8}
\contentsline {subsection}{\numberline {12.8.1}\leavevmode {\color {Chapter }SetName}}{166}{subsection.12.8.1}
\contentsline {subsection}{\numberline {12.8.2}\leavevmode {\color {Chapter }Name}}{166}{subsection.12.8.2}
\contentsline {subsection}{\numberline {12.8.3}\leavevmode {\color {Chapter }InfoText}}{167}{subsection.12.8.3}
\contentsline {subsection}{\numberline {12.8.4}\leavevmode {\color {Chapter }IsInternallyConsistent}}{167}{subsection.12.8.4}
\contentsline {subsection}{\numberline {12.8.5}\leavevmode {\color {Chapter }MemoryUsage}}{167}{subsection.12.8.5}
\contentsline {chapter}{\numberline {13}\leavevmode {\color {Chapter }Types of Objects}}{168}{chapter.13}
\contentsline {section}{\numberline {13.1}\leavevmode {\color {Chapter }Families}}{168}{section.13.1}
\contentsline {subsection}{\numberline {13.1.1}\leavevmode {\color {Chapter }FamilyObj}}{170}{subsection.13.1.1}
\contentsline {subsection}{\numberline {13.1.2}\leavevmode {\color {Chapter }NewFamily}}{170}{subsection.13.1.2}
\contentsline {section}{\numberline {13.2}\leavevmode {\color {Chapter }Filters}}{171}{section.13.2}
\contentsline {subsection}{\numberline {13.2.1}\leavevmode {\color {Chapter }RankFilter}}{171}{subsection.13.2.1}
\contentsline {subsection}{\numberline {13.2.2}\leavevmode {\color {Chapter }NamesFilter}}{171}{subsection.13.2.2}
\contentsline {subsection}{\numberline {13.2.3}\leavevmode {\color {Chapter }FilterByName}}{172}{subsection.13.2.3}
\contentsline {subsection}{\numberline {13.2.4}\leavevmode {\color {Chapter }ShowImpliedFilters}}{172}{subsection.13.2.4}
\contentsline {subsection}{\numberline {13.2.5}\leavevmode {\color {Chapter }FiltersType}}{173}{subsection.13.2.5}
\contentsline {section}{\numberline {13.3}\leavevmode {\color {Chapter }Categories}}{173}{section.13.3}
\contentsline {subsection}{\numberline {13.3.1}\leavevmode {\color {Chapter }IsCategory}}{175}{subsection.13.3.1}
\contentsline {subsection}{\numberline {13.3.2}\leavevmode {\color {Chapter }CategoriesOfObject}}{175}{subsection.13.3.2}
\contentsline {subsection}{\numberline {13.3.3}\leavevmode {\color {Chapter }CategoryByName}}{175}{subsection.13.3.3}
\contentsline {subsection}{\numberline {13.3.4}\leavevmode {\color {Chapter }NewCategory}}{175}{subsection.13.3.4}
\contentsline {subsection}{\numberline {13.3.5}\leavevmode {\color {Chapter }DeclareCategory}}{175}{subsection.13.3.5}
\contentsline {subsection}{\numberline {13.3.6}\leavevmode {\color {Chapter }CategoryFamily}}{176}{subsection.13.3.6}
\contentsline {section}{\numberline {13.4}\leavevmode {\color {Chapter }Representation}}{176}{section.13.4}
\contentsline {subsection}{\numberline {13.4.1}\leavevmode {\color {Chapter }Basic Representations of Objects}}{176}{subsection.13.4.1}
\contentsline {subsection}{\numberline {13.4.2}\leavevmode {\color {Chapter }IsRepresentation}}{177}{subsection.13.4.2}
\contentsline {subsection}{\numberline {13.4.3}\leavevmode {\color {Chapter }RepresentationsOfObject}}{177}{subsection.13.4.3}
\contentsline {subsection}{\numberline {13.4.4}\leavevmode {\color {Chapter }NewRepresentation}}{177}{subsection.13.4.4}
\contentsline {subsection}{\numberline {13.4.5}\leavevmode {\color {Chapter }DeclareRepresentation}}{178}{subsection.13.4.5}
\contentsline {section}{\numberline {13.5}\leavevmode {\color {Chapter }Attributes}}{178}{section.13.5}
\contentsline {subsection}{\numberline {13.5.1}\leavevmode {\color {Chapter }IsAttribute}}{178}{subsection.13.5.1}
\contentsline {subsection}{\numberline {13.5.2}\leavevmode {\color {Chapter }KnownAttributesOfObject}}{178}{subsection.13.5.2}
\contentsline {subsection}{\numberline {13.5.3}\leavevmode {\color {Chapter }NewAttribute}}{179}{subsection.13.5.3}
\contentsline {subsection}{\numberline {13.5.4}\leavevmode {\color {Chapter }DeclareAttribute}}{179}{subsection.13.5.4}
\contentsline {subsection}{\numberline {13.5.5}\leavevmode {\color {Chapter }IsAttributeStoringRep}}{180}{subsection.13.5.5}
\contentsline {section}{\numberline {13.6}\leavevmode {\color {Chapter }Setter and Tester for Attributes}}{180}{section.13.6}
\contentsline {subsection}{\numberline {13.6.1}\leavevmode {\color {Chapter }Tester}}{180}{subsection.13.6.1}
\contentsline {subsection}{\numberline {13.6.2}\leavevmode {\color {Chapter }Setter}}{181}{subsection.13.6.2}
\contentsline {subsection}{\numberline {13.6.3}\leavevmode {\color {Chapter }AttributeValueNotSet}}{181}{subsection.13.6.3}
\contentsline {subsection}{\numberline {13.6.4}\leavevmode {\color {Chapter }InfoAttributes}}{182}{subsection.13.6.4}
\contentsline {subsection}{\numberline {13.6.5}\leavevmode {\color {Chapter }DisableAttributeValueStoring}}{182}{subsection.13.6.5}
\contentsline {subsection}{\numberline {13.6.6}\leavevmode {\color {Chapter }EnableAttributeValueStoring}}{182}{subsection.13.6.6}
\contentsline {section}{\numberline {13.7}\leavevmode {\color {Chapter }Properties}}{183}{section.13.7}
\contentsline {subsection}{\numberline {13.7.1}\leavevmode {\color {Chapter }IsProperty}}{184}{subsection.13.7.1}
\contentsline {subsection}{\numberline {13.7.2}\leavevmode {\color {Chapter }KnownPropertiesOfObject}}{184}{subsection.13.7.2}
\contentsline {subsection}{\numberline {13.7.3}\leavevmode {\color {Chapter }KnownTruePropertiesOfObject}}{184}{subsection.13.7.3}
\contentsline {subsection}{\numberline {13.7.4}\leavevmode {\color {Chapter }NewProperty}}{185}{subsection.13.7.4}
\contentsline {subsection}{\numberline {13.7.5}\leavevmode {\color {Chapter }DeclareProperty}}{185}{subsection.13.7.5}
\contentsline {section}{\numberline {13.8}\leavevmode {\color {Chapter }Other Filters}}{185}{section.13.8}
\contentsline {subsection}{\numberline {13.8.1}\leavevmode {\color {Chapter }NewFilter}}{186}{subsection.13.8.1}
\contentsline {subsection}{\numberline {13.8.2}\leavevmode {\color {Chapter }DeclareFilter}}{186}{subsection.13.8.2}
\contentsline {subsection}{\numberline {13.8.3}\leavevmode {\color {Chapter }SetFilterObj}}{186}{subsection.13.8.3}
\contentsline {subsection}{\numberline {13.8.4}\leavevmode {\color {Chapter }ResetFilterObj}}{186}{subsection.13.8.4}
\contentsline {section}{\numberline {13.9}\leavevmode {\color {Chapter }Types}}{186}{section.13.9}
\contentsline {subsection}{\numberline {13.9.1}\leavevmode {\color {Chapter }TypeObj}}{186}{subsection.13.9.1}
\contentsline {subsection}{\numberline {13.9.2}\leavevmode {\color {Chapter }DataType}}{186}{subsection.13.9.2}
\contentsline {subsection}{\numberline {13.9.3}\leavevmode {\color {Chapter }NewType}}{187}{subsection.13.9.3}
\contentsline {chapter}{\numberline {14}\leavevmode {\color {Chapter }Integers}}{188}{chapter.14}
\contentsline {section}{\numberline {14.1}\leavevmode {\color {Chapter }Integers: Global Variables}}{188}{section.14.1}
\contentsline {subsection}{\numberline {14.1.1}\leavevmode {\color {Chapter }Integers (global variable)}}{188}{subsection.14.1.1}
\contentsline {subsection}{\numberline {14.1.2}\leavevmode {\color {Chapter }IsIntegers}}{189}{subsection.14.1.2}
\contentsline {section}{\numberline {14.2}\leavevmode {\color {Chapter }Elementary Operations for Integers}}{189}{section.14.2}
\contentsline {subsection}{\numberline {14.2.1}\leavevmode {\color {Chapter }IsInt}}{189}{subsection.14.2.1}
\contentsline {subsection}{\numberline {14.2.2}\leavevmode {\color {Chapter }IsPosInt}}{189}{subsection.14.2.2}
\contentsline {subsection}{\numberline {14.2.3}\leavevmode {\color {Chapter }Int}}{189}{subsection.14.2.3}
\contentsline {subsection}{\numberline {14.2.4}\leavevmode {\color {Chapter }IsEvenInt}}{190}{subsection.14.2.4}
\contentsline {subsection}{\numberline {14.2.5}\leavevmode {\color {Chapter }IsOddInt}}{190}{subsection.14.2.5}
\contentsline {subsection}{\numberline {14.2.6}\leavevmode {\color {Chapter }AbsInt}}{190}{subsection.14.2.6}
\contentsline {subsection}{\numberline {14.2.7}\leavevmode {\color {Chapter }SignInt}}{190}{subsection.14.2.7}
\contentsline {subsection}{\numberline {14.2.8}\leavevmode {\color {Chapter }LogInt}}{191}{subsection.14.2.8}
\contentsline {subsection}{\numberline {14.2.9}\leavevmode {\color {Chapter }RootInt}}{191}{subsection.14.2.9}
\contentsline {subsection}{\numberline {14.2.10}\leavevmode {\color {Chapter }SmallestRootInt}}{191}{subsection.14.2.10}
\contentsline {subsection}{\numberline {14.2.11}\leavevmode {\color {Chapter }ListOfDigits}}{192}{subsection.14.2.11}
\contentsline {subsection}{\numberline {14.2.12}\leavevmode {\color {Chapter }Random (for integers)}}{192}{subsection.14.2.12}
\contentsline {section}{\numberline {14.3}\leavevmode {\color {Chapter }Quotients and Remainders}}{192}{section.14.3}
\contentsline {subsection}{\numberline {14.3.1}\leavevmode {\color {Chapter }QuoInt}}{192}{subsection.14.3.1}
\contentsline {subsection}{\numberline {14.3.2}\leavevmode {\color {Chapter }BestQuoInt}}{192}{subsection.14.3.2}
\contentsline {subsection}{\numberline {14.3.3}\leavevmode {\color {Chapter }RemInt}}{193}{subsection.14.3.3}
\contentsline {subsection}{\numberline {14.3.4}\leavevmode {\color {Chapter }GcdInt}}{193}{subsection.14.3.4}
\contentsline {subsection}{\numberline {14.3.5}\leavevmode {\color {Chapter }Gcdex}}{193}{subsection.14.3.5}
\contentsline {subsection}{\numberline {14.3.6}\leavevmode {\color {Chapter }LcmInt}}{194}{subsection.14.3.6}
\contentsline {subsection}{\numberline {14.3.7}\leavevmode {\color {Chapter }CoefficientsQadic}}{194}{subsection.14.3.7}
\contentsline {subsection}{\numberline {14.3.8}\leavevmode {\color {Chapter }CoefficientsMultiadic}}{194}{subsection.14.3.8}
\contentsline {subsection}{\numberline {14.3.9}\leavevmode {\color {Chapter }ChineseRem}}{195}{subsection.14.3.9}
\contentsline {subsection}{\numberline {14.3.10}\leavevmode {\color {Chapter }PowerModInt}}{195}{subsection.14.3.10}
\contentsline {section}{\numberline {14.4}\leavevmode {\color {Chapter }Prime Integers and Factorization}}{195}{section.14.4}
\contentsline {subsection}{\numberline {14.4.1}\leavevmode {\color {Chapter }Primes}}{195}{subsection.14.4.1}
\contentsline {subsection}{\numberline {14.4.2}\leavevmode {\color {Chapter }IsPrimeInt}}{196}{subsection.14.4.2}
\contentsline {subsection}{\numberline {14.4.3}\leavevmode {\color {Chapter }PrimalityProof}}{196}{subsection.14.4.3}
\contentsline {subsection}{\numberline {14.4.4}\leavevmode {\color {Chapter }IsPrimePowerInt}}{197}{subsection.14.4.4}
\contentsline {subsection}{\numberline {14.4.5}\leavevmode {\color {Chapter }NextPrimeInt}}{197}{subsection.14.4.5}
\contentsline {subsection}{\numberline {14.4.6}\leavevmode {\color {Chapter }PrevPrimeInt}}{197}{subsection.14.4.6}
\contentsline {subsection}{\numberline {14.4.7}\leavevmode {\color {Chapter }FactorsInt}}{197}{subsection.14.4.7}
\contentsline {subsection}{\numberline {14.4.8}\leavevmode {\color {Chapter }PrimeDivisors}}{198}{subsection.14.4.8}
\contentsline {subsection}{\numberline {14.4.9}\leavevmode {\color {Chapter }PartialFactorization}}{199}{subsection.14.4.9}
\contentsline {subsection}{\numberline {14.4.10}\leavevmode {\color {Chapter }PrintFactorsInt}}{200}{subsection.14.4.10}
\contentsline {subsection}{\numberline {14.4.11}\leavevmode {\color {Chapter }PrimePowersInt}}{200}{subsection.14.4.11}
\contentsline {subsection}{\numberline {14.4.12}\leavevmode {\color {Chapter }DivisorsInt}}{200}{subsection.14.4.12}
\contentsline {section}{\numberline {14.5}\leavevmode {\color {Chapter }Residue Class Rings}}{200}{section.14.5}
\contentsline {subsection}{\numberline {14.5.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }mod (for residue class rings)}}{200}{subsection.14.5.1}
\contentsline {subsection}{\numberline {14.5.2}\leavevmode {\color {Chapter }ZmodnZ}}{201}{subsection.14.5.2}
\contentsline {subsection}{\numberline {14.5.3}\leavevmode {\color {Chapter }ZmodnZObj (for a residue class family and integer)}}{201}{subsection.14.5.3}
\contentsline {subsection}{\numberline {14.5.4}\leavevmode {\color {Chapter }IsZmodnZObj}}{201}{subsection.14.5.4}
\contentsline {section}{\numberline {14.6}\leavevmode {\color {Chapter }Check Digits}}{202}{section.14.6}
\contentsline {subsection}{\numberline {14.6.1}\leavevmode {\color {Chapter }CheckDigitISBN}}{202}{subsection.14.6.1}
\contentsline {subsection}{\numberline {14.6.2}\leavevmode {\color {Chapter }CheckDigitTestFunction}}{203}{subsection.14.6.2}
\contentsline {section}{\numberline {14.7}\leavevmode {\color {Chapter }Random Sources}}{203}{section.14.7}
\contentsline {subsection}{\numberline {14.7.1}\leavevmode {\color {Chapter }IsRandomSource}}{203}{subsection.14.7.1}
\contentsline {subsection}{\numberline {14.7.2}\leavevmode {\color {Chapter }Random (for random source and list)}}{204}{subsection.14.7.2}
\contentsline {subsection}{\numberline {14.7.3}\leavevmode {\color {Chapter }State}}{204}{subsection.14.7.3}
\contentsline {subsection}{\numberline {14.7.4}\leavevmode {\color {Chapter }IsMersenneTwister}}{205}{subsection.14.7.4}
\contentsline {subsection}{\numberline {14.7.5}\leavevmode {\color {Chapter }RandomSource}}{205}{subsection.14.7.5}
\contentsline {section}{\numberline {14.8}\leavevmode {\color {Chapter }Bitfields}}{206}{section.14.8}
\contentsline {subsection}{\numberline {14.8.1}\leavevmode {\color {Chapter }MakeBitfields}}{206}{subsection.14.8.1}
\contentsline {subsection}{\numberline {14.8.2}\leavevmode {\color {Chapter }BuildBitfields}}{206}{subsection.14.8.2}
\contentsline {chapter}{\numberline {15}\leavevmode {\color {Chapter }Number Theory}}{208}{chapter.15}
\contentsline {section}{\numberline {15.1}\leavevmode {\color {Chapter }InfoNumtheor (Info Class)}}{208}{section.15.1}
\contentsline {subsection}{\numberline {15.1.1}\leavevmode {\color {Chapter }InfoNumtheor}}{208}{subsection.15.1.1}
\contentsline {section}{\numberline {15.2}\leavevmode {\color {Chapter }Prime Residues}}{208}{section.15.2}
\contentsline {subsection}{\numberline {15.2.1}\leavevmode {\color {Chapter }PrimeResidues}}{208}{subsection.15.2.1}
\contentsline {subsection}{\numberline {15.2.2}\leavevmode {\color {Chapter }Phi}}{209}{subsection.15.2.2}
\contentsline {subsection}{\numberline {15.2.3}\leavevmode {\color {Chapter }Lambda}}{209}{subsection.15.2.3}
\contentsline {subsection}{\numberline {15.2.4}\leavevmode {\color {Chapter }GeneratorsPrimeResidues}}{209}{subsection.15.2.4}
\contentsline {section}{\numberline {15.3}\leavevmode {\color {Chapter }Primitive Roots and Discrete Logarithms}}{210}{section.15.3}
\contentsline {subsection}{\numberline {15.3.1}\leavevmode {\color {Chapter }OrderMod}}{210}{subsection.15.3.1}
\contentsline {subsection}{\numberline {15.3.2}\leavevmode {\color {Chapter }LogMod}}{210}{subsection.15.3.2}
\contentsline {subsection}{\numberline {15.3.3}\leavevmode {\color {Chapter }PrimitiveRootMod}}{211}{subsection.15.3.3}
\contentsline {subsection}{\numberline {15.3.4}\leavevmode {\color {Chapter }IsPrimitiveRootMod}}{211}{subsection.15.3.4}
\contentsline {section}{\numberline {15.4}\leavevmode {\color {Chapter }Roots Modulo Integers}}{211}{section.15.4}
\contentsline {subsection}{\numberline {15.4.1}\leavevmode {\color {Chapter }Jacobi}}{211}{subsection.15.4.1}
\contentsline {subsection}{\numberline {15.4.2}\leavevmode {\color {Chapter }Legendre}}{212}{subsection.15.4.2}
\contentsline {subsection}{\numberline {15.4.3}\leavevmode {\color {Chapter }RootMod}}{212}{subsection.15.4.3}
\contentsline {subsection}{\numberline {15.4.4}\leavevmode {\color {Chapter }RootsMod}}{213}{subsection.15.4.4}
\contentsline {subsection}{\numberline {15.4.5}\leavevmode {\color {Chapter }RootsUnityMod}}{213}{subsection.15.4.5}
\contentsline {section}{\numberline {15.5}\leavevmode {\color {Chapter }Multiplicative Arithmetic Functions}}{214}{section.15.5}
\contentsline {subsection}{\numberline {15.5.1}\leavevmode {\color {Chapter }Sigma}}{214}{subsection.15.5.1}
\contentsline {subsection}{\numberline {15.5.2}\leavevmode {\color {Chapter }Tau}}{214}{subsection.15.5.2}
\contentsline {subsection}{\numberline {15.5.3}\leavevmode {\color {Chapter }MoebiusMu}}{215}{subsection.15.5.3}
\contentsline {section}{\numberline {15.6}\leavevmode {\color {Chapter }Continued Fractions}}{215}{section.15.6}
\contentsline {subsection}{\numberline {15.6.1}\leavevmode {\color {Chapter }ContinuedFractionExpansionOfRoot}}{215}{subsection.15.6.1}
\contentsline {subsection}{\numberline {15.6.2}\leavevmode {\color {Chapter }ContinuedFractionApproximationOfRoot}}{215}{subsection.15.6.2}
\contentsline {section}{\numberline {15.7}\leavevmode {\color {Chapter }Miscellaneous}}{216}{section.15.7}
\contentsline {subsection}{\numberline {15.7.1}\leavevmode {\color {Chapter }PValuation}}{216}{subsection.15.7.1}
\contentsline {subsection}{\numberline {15.7.2}\leavevmode {\color {Chapter }TwoSquares}}{216}{subsection.15.7.2}
\contentsline {chapter}{\numberline {16}\leavevmode {\color {Chapter }Combinatorics}}{218}{chapter.16}
\contentsline {section}{\numberline {16.1}\leavevmode {\color {Chapter }Combinatorial Numbers}}{218}{section.16.1}
\contentsline {subsection}{\numberline {16.1.1}\leavevmode {\color {Chapter }Factorial}}{218}{subsection.16.1.1}
\contentsline {subsection}{\numberline {16.1.2}\leavevmode {\color {Chapter }Binomial}}{218}{subsection.16.1.2}
\contentsline {subsection}{\numberline {16.1.3}\leavevmode {\color {Chapter }Bell}}{219}{subsection.16.1.3}
\contentsline {subsection}{\numberline {16.1.4}\leavevmode {\color {Chapter }Bernoulli}}{219}{subsection.16.1.4}
\contentsline {subsection}{\numberline {16.1.5}\leavevmode {\color {Chapter }Stirling1}}{220}{subsection.16.1.5}
\contentsline {subsection}{\numberline {16.1.6}\leavevmode {\color {Chapter }Stirling2}}{220}{subsection.16.1.6}
\contentsline {section}{\numberline {16.2}\leavevmode {\color {Chapter }Combinations, Arrangements and Tuples}}{221}{section.16.2}
\contentsline {subsection}{\numberline {16.2.1}\leavevmode {\color {Chapter }Combinations}}{221}{subsection.16.2.1}
\contentsline {subsection}{\numberline {16.2.2}\leavevmode {\color {Chapter }Iterator and enumerator of combinations}}{221}{subsection.16.2.2}
\contentsline {subsection}{\numberline {16.2.3}\leavevmode {\color {Chapter }NrCombinations}}{222}{subsection.16.2.3}
\contentsline {subsection}{\numberline {16.2.4}\leavevmode {\color {Chapter }Arrangements}}{222}{subsection.16.2.4}
\contentsline {subsection}{\numberline {16.2.5}\leavevmode {\color {Chapter }NrArrangements}}{222}{subsection.16.2.5}
\contentsline {subsection}{\numberline {16.2.6}\leavevmode {\color {Chapter }UnorderedTuples}}{223}{subsection.16.2.6}
\contentsline {subsection}{\numberline {16.2.7}\leavevmode {\color {Chapter }NrUnorderedTuples}}{223}{subsection.16.2.7}
\contentsline {subsection}{\numberline {16.2.8}\leavevmode {\color {Chapter }Tuples}}{223}{subsection.16.2.8}
\contentsline {subsection}{\numberline {16.2.9}\leavevmode {\color {Chapter }EnumeratorOfTuples}}{224}{subsection.16.2.9}
\contentsline {subsection}{\numberline {16.2.10}\leavevmode {\color {Chapter }IteratorOfTuples}}{224}{subsection.16.2.10}
\contentsline {subsection}{\numberline {16.2.11}\leavevmode {\color {Chapter }NrTuples}}{224}{subsection.16.2.11}
\contentsline {subsection}{\numberline {16.2.12}\leavevmode {\color {Chapter }PermutationsList}}{224}{subsection.16.2.12}
\contentsline {subsection}{\numberline {16.2.13}\leavevmode {\color {Chapter }NrPermutationsList}}{225}{subsection.16.2.13}
\contentsline {subsection}{\numberline {16.2.14}\leavevmode {\color {Chapter }Derangements}}{225}{subsection.16.2.14}
\contentsline {subsection}{\numberline {16.2.15}\leavevmode {\color {Chapter }NrDerangements}}{225}{subsection.16.2.15}
\contentsline {subsection}{\numberline {16.2.16}\leavevmode {\color {Chapter }PartitionsSet}}{226}{subsection.16.2.16}
\contentsline {subsection}{\numberline {16.2.17}\leavevmode {\color {Chapter }NrPartitionsSet}}{226}{subsection.16.2.17}
\contentsline {subsection}{\numberline {16.2.18}\leavevmode {\color {Chapter }Partitions}}{227}{subsection.16.2.18}
\contentsline {subsection}{\numberline {16.2.19}\leavevmode {\color {Chapter }IteratorOfPartitions}}{227}{subsection.16.2.19}
\contentsline {subsection}{\numberline {16.2.20}\leavevmode {\color {Chapter }NrPartitions}}{227}{subsection.16.2.20}
\contentsline {subsection}{\numberline {16.2.21}\leavevmode {\color {Chapter }OrderedPartitions}}{227}{subsection.16.2.21}
\contentsline {subsection}{\numberline {16.2.22}\leavevmode {\color {Chapter }NrOrderedPartitions}}{228}{subsection.16.2.22}
\contentsline {subsection}{\numberline {16.2.23}\leavevmode {\color {Chapter }PartitionsGreatestLE}}{228}{subsection.16.2.23}
\contentsline {subsection}{\numberline {16.2.24}\leavevmode {\color {Chapter }PartitionsGreatestEQ}}{228}{subsection.16.2.24}
\contentsline {subsection}{\numberline {16.2.25}\leavevmode {\color {Chapter }RestrictedPartitions}}{228}{subsection.16.2.25}
\contentsline {subsection}{\numberline {16.2.26}\leavevmode {\color {Chapter }NrRestrictedPartitions}}{229}{subsection.16.2.26}
\contentsline {subsection}{\numberline {16.2.27}\leavevmode {\color {Chapter }SignPartition}}{229}{subsection.16.2.27}
\contentsline {subsection}{\numberline {16.2.28}\leavevmode {\color {Chapter }AssociatedPartition}}{229}{subsection.16.2.28}
\contentsline {subsection}{\numberline {16.2.29}\leavevmode {\color {Chapter }PowerPartition}}{230}{subsection.16.2.29}
\contentsline {subsection}{\numberline {16.2.30}\leavevmode {\color {Chapter }PartitionTuples}}{230}{subsection.16.2.30}
\contentsline {subsection}{\numberline {16.2.31}\leavevmode {\color {Chapter }NrPartitionTuples}}{230}{subsection.16.2.31}
\contentsline {subsection}{\numberline {16.2.32}\leavevmode {\color {Chapter }BetaSet}}{230}{subsection.16.2.32}
\contentsline {section}{\numberline {16.3}\leavevmode {\color {Chapter }Fibonacci and Lucas Sequences}}{231}{section.16.3}
\contentsline {subsection}{\numberline {16.3.1}\leavevmode {\color {Chapter }Fibonacci}}{231}{subsection.16.3.1}
\contentsline {subsection}{\numberline {16.3.2}\leavevmode {\color {Chapter }Lucas}}{231}{subsection.16.3.2}
\contentsline {section}{\numberline {16.4}\leavevmode {\color {Chapter }Permanent of a Matrix}}{232}{section.16.4}
\contentsline {subsection}{\numberline {16.4.1}\leavevmode {\color {Chapter }Permanent}}{232}{subsection.16.4.1}
\contentsline {chapter}{\numberline {17}\leavevmode {\color {Chapter }Rational Numbers}}{233}{chapter.17}
\contentsline {section}{\numberline {17.1}\leavevmode {\color {Chapter }Rationals: Global Variables}}{233}{section.17.1}
\contentsline {subsection}{\numberline {17.1.1}\leavevmode {\color {Chapter }Rationals}}{233}{subsection.17.1.1}
\contentsline {section}{\numberline {17.2}\leavevmode {\color {Chapter }Elementary Operations for Rationals}}{234}{section.17.2}
\contentsline {subsection}{\numberline {17.2.1}\leavevmode {\color {Chapter }IsRat}}{234}{subsection.17.2.1}
\contentsline {subsection}{\numberline {17.2.2}\leavevmode {\color {Chapter }IsPosRat}}{234}{subsection.17.2.2}
\contentsline {subsection}{\numberline {17.2.3}\leavevmode {\color {Chapter }IsNegRat}}{234}{subsection.17.2.3}
\contentsline {subsection}{\numberline {17.2.4}\leavevmode {\color {Chapter }NumeratorRat}}{234}{subsection.17.2.4}
\contentsline {subsection}{\numberline {17.2.5}\leavevmode {\color {Chapter }DenominatorRat}}{235}{subsection.17.2.5}
\contentsline {subsection}{\numberline {17.2.6}\leavevmode {\color {Chapter }Rat}}{235}{subsection.17.2.6}
\contentsline {subsection}{\numberline {17.2.7}\leavevmode {\color {Chapter }Random (for rationals)}}{235}{subsection.17.2.7}
\contentsline {chapter}{\numberline {18}\leavevmode {\color {Chapter }Cyclotomic Numbers}}{236}{chapter.18}
\contentsline {section}{\numberline {18.1}\leavevmode {\color {Chapter }Operations for Cyclotomics}}{236}{section.18.1}
\contentsline {subsection}{\numberline {18.1.1}\leavevmode {\color {Chapter }E}}{236}{subsection.18.1.1}
\contentsline {subsection}{\numberline {18.1.2}\leavevmode {\color {Chapter }Cyclotomics}}{236}{subsection.18.1.2}
\contentsline {subsection}{\numberline {18.1.3}\leavevmode {\color {Chapter }IsCyclotomic}}{237}{subsection.18.1.3}
\contentsline {subsection}{\numberline {18.1.4}\leavevmode {\color {Chapter }IsIntegralCyclotomic}}{237}{subsection.18.1.4}
\contentsline {subsection}{\numberline {18.1.5}\leavevmode {\color {Chapter }Int (for a cyclotomic)}}{238}{subsection.18.1.5}
\contentsline {subsection}{\numberline {18.1.6}\leavevmode {\color {Chapter }String (for a cyclotomic)}}{238}{subsection.18.1.6}
\contentsline {subsection}{\numberline {18.1.7}\leavevmode {\color {Chapter }Conductor (for a cyclotomic)}}{238}{subsection.18.1.7}
\contentsline {subsection}{\numberline {18.1.8}\leavevmode {\color {Chapter }AbsoluteValue}}{239}{subsection.18.1.8}
\contentsline {subsection}{\numberline {18.1.9}\leavevmode {\color {Chapter }RoundCyc}}{239}{subsection.18.1.9}
\contentsline {subsection}{\numberline {18.1.10}\leavevmode {\color {Chapter }CoeffsCyc}}{239}{subsection.18.1.10}
\contentsline {subsection}{\numberline {18.1.11}\leavevmode {\color {Chapter }DenominatorCyc}}{240}{subsection.18.1.11}
\contentsline {subsection}{\numberline {18.1.12}\leavevmode {\color {Chapter }ExtRepOfObj (for a cyclotomic)}}{240}{subsection.18.1.12}
\contentsline {subsection}{\numberline {18.1.13}\leavevmode {\color {Chapter }DescriptionOfRootOfUnity}}{240}{subsection.18.1.13}
\contentsline {subsection}{\numberline {18.1.14}\leavevmode {\color {Chapter }IsGaussInt}}{240}{subsection.18.1.14}
\contentsline {subsection}{\numberline {18.1.15}\leavevmode {\color {Chapter }IsGaussRat}}{240}{subsection.18.1.15}
\contentsline {subsection}{\numberline {18.1.16}\leavevmode {\color {Chapter }DefaultField (for cyclotomics)}}{241}{subsection.18.1.16}
\contentsline {section}{\numberline {18.2}\leavevmode {\color {Chapter }Infinity and negative Infinity}}{241}{section.18.2}
\contentsline {subsection}{\numberline {18.2.1}\leavevmode {\color {Chapter }IsInfinity}}{241}{subsection.18.2.1}
\contentsline {section}{\numberline {18.3}\leavevmode {\color {Chapter }Comparisons of Cyclotomics}}{242}{section.18.3}
\contentsline {section}{\numberline {18.4}\leavevmode {\color {Chapter }ATLAS Irrationalities}}{242}{section.18.4}
\contentsline {subsection}{\numberline {18.4.1}\leavevmode {\color {Chapter }EB, EC, $\dots $, EH}}{242}{subsection.18.4.1}
\contentsline {subsection}{\numberline {18.4.2}\leavevmode {\color {Chapter }EI and ER}}{243}{subsection.18.4.2}
\contentsline {subsection}{\numberline {18.4.3}\leavevmode {\color {Chapter }EY, EX, $\dots $, ES}}{243}{subsection.18.4.3}
\contentsline {subsection}{\numberline {18.4.4}\leavevmode {\color {Chapter }EM, EL, $\dots $, EJ}}{244}{subsection.18.4.4}
\contentsline {subsection}{\numberline {18.4.5}\leavevmode {\color {Chapter }NK}}{244}{subsection.18.4.5}
\contentsline {subsection}{\numberline {18.4.6}\leavevmode {\color {Chapter }AtlasIrrationality}}{245}{subsection.18.4.6}
\contentsline {section}{\numberline {18.5}\leavevmode {\color {Chapter }Galois Conjugacy of Cyclotomics}}{245}{section.18.5}
\contentsline {subsection}{\numberline {18.5.1}\leavevmode {\color {Chapter }GaloisCyc (for a cyclotomic)}}{245}{subsection.18.5.1}
\contentsline {subsection}{\numberline {18.5.2}\leavevmode {\color {Chapter }ComplexConjugate}}{246}{subsection.18.5.2}
\contentsline {subsection}{\numberline {18.5.3}\leavevmode {\color {Chapter }StarCyc}}{246}{subsection.18.5.3}
\contentsline {subsection}{\numberline {18.5.4}\leavevmode {\color {Chapter }Quadratic}}{246}{subsection.18.5.4}
\contentsline {subsection}{\numberline {18.5.5}\leavevmode {\color {Chapter }GaloisMat}}{247}{subsection.18.5.5}
\contentsline {subsection}{\numberline {18.5.6}\leavevmode {\color {Chapter }RationalizedMat}}{248}{subsection.18.5.6}
\contentsline {section}{\numberline {18.6}\leavevmode {\color {Chapter }Internally Represented Cyclotomics}}{248}{section.18.6}
\contentsline {subsection}{\numberline {18.6.1}\leavevmode {\color {Chapter }SetCyclotomicsLimit}}{249}{subsection.18.6.1}
\contentsline {chapter}{\numberline {19}\leavevmode {\color {Chapter }Floats}}{250}{chapter.19}
\contentsline {section}{\numberline {19.1}\leavevmode {\color {Chapter }A sample run}}{250}{section.19.1}
\contentsline {section}{\numberline {19.2}\leavevmode {\color {Chapter }Methods}}{251}{section.19.2}
\contentsline {subsection}{\numberline {19.2.1}\leavevmode {\color {Chapter }Float creators}}{252}{subsection.19.2.1}
\contentsline {subsection}{\numberline {19.2.2}\leavevmode {\color {Chapter }Rat (for floats)}}{252}{subsection.19.2.2}
\contentsline {subsection}{\numberline {19.2.3}\leavevmode {\color {Chapter }Cyc (for floats)}}{253}{subsection.19.2.3}
\contentsline {subsection}{\numberline {19.2.4}\leavevmode {\color {Chapter }SetFloats}}{253}{subsection.19.2.4}
\contentsline {subsection}{\numberline {19.2.5}\leavevmode {\color {Chapter }FLOAT (constants)}}{253}{subsection.19.2.5}
\contentsline {subsection}{\numberline {19.2.6}\leavevmode {\color {Chapter }EqFloat}}{254}{subsection.19.2.6}
\contentsline {subsection}{\numberline {19.2.7}\leavevmode {\color {Chapter }PrecisionFloat}}{254}{subsection.19.2.7}
\contentsline {subsection}{\numberline {19.2.8}\leavevmode {\color {Chapter }SignBit}}{254}{subsection.19.2.8}
\contentsline {subsection}{\numberline {19.2.9}\leavevmode {\color {Chapter }Infinity testers}}{254}{subsection.19.2.9}
\contentsline {subsection}{\numberline {19.2.10}\leavevmode {\color {Chapter }Standard mathematical operations}}{255}{subsection.19.2.10}
\contentsline {section}{\numberline {19.3}\leavevmode {\color {Chapter }High-precision-specific methods}}{256}{section.19.3}
\contentsline {section}{\numberline {19.4}\leavevmode {\color {Chapter }Complex arithmetic}}{256}{section.19.4}
\contentsline {subsection}{\numberline {19.4.1}\leavevmode {\color {Chapter }Argument (for complex floats)}}{256}{subsection.19.4.1}
\contentsline {section}{\numberline {19.5}\leavevmode {\color {Chapter }Interval-specific methods}}{257}{section.19.5}
\contentsline {subsection}{\numberline {19.5.1}\leavevmode {\color {Chapter }Sup}}{257}{subsection.19.5.1}
\contentsline {subsection}{\numberline {19.5.2}\leavevmode {\color {Chapter }Inf}}{257}{subsection.19.5.2}
\contentsline {subsection}{\numberline {19.5.3}\leavevmode {\color {Chapter }Mid}}{257}{subsection.19.5.3}
\contentsline {subsection}{\numberline {19.5.4}\leavevmode {\color {Chapter }AbsoluteDiameter}}{257}{subsection.19.5.4}
\contentsline {subsection}{\numberline {19.5.5}\leavevmode {\color {Chapter }RelativeDiameter}}{257}{subsection.19.5.5}
\contentsline {subsection}{\numberline {19.5.6}\leavevmode {\color {Chapter }IsDisjoint}}{257}{subsection.19.5.6}
\contentsline {subsection}{\numberline {19.5.7}\leavevmode {\color {Chapter }IsSubset (for interval floats)}}{258}{subsection.19.5.7}
\contentsline {subsection}{\numberline {19.5.8}\leavevmode {\color {Chapter }IncreaseInterval}}{258}{subsection.19.5.8}
\contentsline {subsection}{\numberline {19.5.9}\leavevmode {\color {Chapter }BlowupInterval}}{258}{subsection.19.5.9}
\contentsline {subsection}{\numberline {19.5.10}\leavevmode {\color {Chapter }BisectInterval}}{258}{subsection.19.5.10}
\contentsline {chapter}{\numberline {20}\leavevmode {\color {Chapter }Booleans}}{259}{chapter.20}
\contentsline {section}{\numberline {20.1}\leavevmode {\color {Chapter }IsBool (Filter)}}{259}{section.20.1}
\contentsline {subsection}{\numberline {20.1.1}\leavevmode {\color {Chapter }IsBool}}{259}{subsection.20.1.1}
\contentsline {section}{\numberline {20.2}\leavevmode {\color {Chapter }Fail (Variable)}}{259}{section.20.2}
\contentsline {subsection}{\numberline {20.2.1}\leavevmode {\color {Chapter }fail}}{259}{subsection.20.2.1}
\contentsline {section}{\numberline {20.3}\leavevmode {\color {Chapter }Comparisons of Booleans}}{260}{section.20.3}
\contentsline {subsection}{\numberline {20.3.1}\leavevmode {\color {Chapter }Equality and inequality of Booleans}}{260}{subsection.20.3.1}
\contentsline {subsection}{\numberline {20.3.2}\leavevmode {\color {Chapter }Ordering of Booleans}}{260}{subsection.20.3.2}
\contentsline {section}{\numberline {20.4}\leavevmode {\color {Chapter }Operations for Booleans}}{260}{section.20.4}
\contentsline {subsection}{\numberline {20.4.1}\leavevmode {\color {Chapter }Logical disjunction}}{260}{subsection.20.4.1}
\contentsline {subsection}{\numberline {20.4.2}\leavevmode {\color {Chapter }Logical conjunction}}{261}{subsection.20.4.2}
\contentsline {subsection}{\numberline {20.4.3}\leavevmode {\color {Chapter }Logical negation}}{261}{subsection.20.4.3}
\contentsline {chapter}{\numberline {21}\leavevmode {\color {Chapter }Lists}}{263}{chapter.21}
\contentsline {section}{\numberline {21.1}\leavevmode {\color {Chapter }List Categories}}{263}{section.21.1}
\contentsline {subsection}{\numberline {21.1.1}\leavevmode {\color {Chapter }IsList}}{263}{subsection.21.1.1}
\contentsline {subsection}{\numberline {21.1.2}\leavevmode {\color {Chapter }IsDenseList}}{264}{subsection.21.1.2}
\contentsline {subsection}{\numberline {21.1.3}\leavevmode {\color {Chapter }IsHomogeneousList}}{264}{subsection.21.1.3}
\contentsline {subsection}{\numberline {21.1.4}\leavevmode {\color {Chapter }IsTable}}{264}{subsection.21.1.4}
\contentsline {subsection}{\numberline {21.1.5}\leavevmode {\color {Chapter }IsRectangularTable}}{265}{subsection.21.1.5}
\contentsline {subsection}{\numberline {21.1.6}\leavevmode {\color {Chapter }IsConstantTimeAccessList}}{265}{subsection.21.1.6}
\contentsline {section}{\numberline {21.2}\leavevmode {\color {Chapter }Basic Operations for Lists}}{265}{section.21.2}
\contentsline {subsection}{\numberline {21.2.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }[\texttt {\char 92\relax }]}}{265}{subsection.21.2.1}
\contentsline {section}{\numberline {21.3}\leavevmode {\color {Chapter }List Elements}}{266}{section.21.3}
\contentsline {subsection}{\numberline {21.3.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }\texttt {\char 123\relax }\texttt {\char 92\relax }\texttt {\char 125\relax }}}{267}{subsection.21.3.1}
\contentsline {section}{\numberline {21.4}\leavevmode {\color {Chapter }List Assignment}}{267}{section.21.4}
\contentsline {subsection}{\numberline {21.4.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }\texttt {\char 123\relax }\texttt {\char 92\relax }\texttt {\char 125\relax }\texttt {\char 92\relax }:\texttt {\char 92\relax }=}}{269}{subsection.21.4.1}
\contentsline {subsection}{\numberline {21.4.2}\leavevmode {\color {Chapter }Add}}{269}{subsection.21.4.2}
\contentsline {subsection}{\numberline {21.4.3}\leavevmode {\color {Chapter }Remove}}{270}{subsection.21.4.3}
\contentsline {subsection}{\numberline {21.4.4}\leavevmode {\color {Chapter }CopyListEntries}}{270}{subsection.21.4.4}
\contentsline {subsection}{\numberline {21.4.5}\leavevmode {\color {Chapter }Append}}{270}{subsection.21.4.5}
\contentsline {section}{\numberline {21.5}\leavevmode {\color {Chapter }IsBound and Unbind for Lists}}{271}{section.21.5}
\contentsline {subsection}{\numberline {21.5.1}\leavevmode {\color {Chapter }IsBound (for a list index)}}{271}{subsection.21.5.1}
\contentsline {subsection}{\numberline {21.5.2}\leavevmode {\color {Chapter }GetWithDefault}}{271}{subsection.21.5.2}
\contentsline {subsection}{\numberline {21.5.3}\leavevmode {\color {Chapter }Unbind (unbind a list entry)}}{272}{subsection.21.5.3}
\contentsline {section}{\numberline {21.6}\leavevmode {\color {Chapter }Identical Lists}}{272}{section.21.6}
\contentsline {section}{\numberline {21.7}\leavevmode {\color {Chapter }Duplication of Lists}}{274}{section.21.7}
\contentsline {section}{\numberline {21.8}\leavevmode {\color {Chapter }Membership Test for Lists}}{275}{section.21.8}
\contentsline {subsection}{\numberline {21.8.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }in (element test for lists)}}{275}{subsection.21.8.1}
\contentsline {section}{\numberline {21.9}\leavevmode {\color {Chapter }Enlarging Internally Represented Lists}}{275}{section.21.9}
\contentsline {subsection}{\numberline {21.9.1}\leavevmode {\color {Chapter }EmptyPlist}}{276}{subsection.21.9.1}
\contentsline {section}{\numberline {21.10}\leavevmode {\color {Chapter }Comparisons of Lists}}{276}{section.21.10}
\contentsline {section}{\numberline {21.11}\leavevmode {\color {Chapter }Arithmetic for Lists}}{277}{section.21.11}
\contentsline {section}{\numberline {21.12}\leavevmode {\color {Chapter }Filters Controlling the Arithmetic Behaviour of Lists}}{278}{section.21.12}
\contentsline {subsection}{\numberline {21.12.1}\leavevmode {\color {Chapter }IsGeneralizedRowVector}}{278}{subsection.21.12.1}
\contentsline {subsection}{\numberline {21.12.2}\leavevmode {\color {Chapter }IsMultiplicativeGeneralizedRowVector}}{278}{subsection.21.12.2}
\contentsline {subsection}{\numberline {21.12.3}\leavevmode {\color {Chapter }IsListDefault}}{279}{subsection.21.12.3}
\contentsline {subsection}{\numberline {21.12.4}\leavevmode {\color {Chapter }NestingDepthA}}{279}{subsection.21.12.4}
\contentsline {subsection}{\numberline {21.12.5}\leavevmode {\color {Chapter }NestingDepthM}}{279}{subsection.21.12.5}
\contentsline {section}{\numberline {21.13}\leavevmode {\color {Chapter }Additive Arithmetic for Lists}}{280}{section.21.13}
\contentsline {subsection}{\numberline {21.13.1}\leavevmode {\color {Chapter }Zero for lists}}{280}{subsection.21.13.1}
\contentsline {subsection}{\numberline {21.13.2}\leavevmode {\color {Chapter }AdditiveInverse for lists}}{280}{subsection.21.13.2}
\contentsline {subsection}{\numberline {21.13.3}\leavevmode {\color {Chapter }Addition of lists}}{280}{subsection.21.13.3}
\contentsline {subsection}{\numberline {21.13.4}\leavevmode {\color {Chapter }Subtraction of lists}}{281}{subsection.21.13.4}
\contentsline {section}{\numberline {21.14}\leavevmode {\color {Chapter }Multiplicative Arithmetic for Lists}}{281}{section.21.14}
\contentsline {subsection}{\numberline {21.14.1}\leavevmode {\color {Chapter }One for lists}}{282}{subsection.21.14.1}
\contentsline {subsection}{\numberline {21.14.2}\leavevmode {\color {Chapter }Inverse for lists}}{282}{subsection.21.14.2}
\contentsline {subsection}{\numberline {21.14.3}\leavevmode {\color {Chapter }Multiplication of lists}}{282}{subsection.21.14.3}
\contentsline {subsection}{\numberline {21.14.4}\leavevmode {\color {Chapter }Division of lists}}{283}{subsection.21.14.4}
\contentsline {subsection}{\numberline {21.14.5}\leavevmode {\color {Chapter }mod for lists}}{284}{subsection.21.14.5}
\contentsline {subsection}{\numberline {21.14.6}\leavevmode {\color {Chapter }Left quotients of lists}}{284}{subsection.21.14.6}
\contentsline {section}{\numberline {21.15}\leavevmode {\color {Chapter }Mutability Status and List Arithmetic}}{284}{section.21.15}
\contentsline {subsection}{\numberline {21.15.1}\leavevmode {\color {Chapter }ListWithIdenticalEntries}}{285}{subsection.21.15.1}
\contentsline {section}{\numberline {21.16}\leavevmode {\color {Chapter }Finding Positions in Lists}}{286}{section.21.16}
\contentsline {subsection}{\numberline {21.16.1}\leavevmode {\color {Chapter }Position}}{286}{subsection.21.16.1}
\contentsline {subsection}{\numberline {21.16.2}\leavevmode {\color {Chapter }Positions}}{286}{subsection.21.16.2}
\contentsline {subsection}{\numberline {21.16.3}\leavevmode {\color {Chapter }PositionCanonical}}{286}{subsection.21.16.3}
\contentsline {subsection}{\numberline {21.16.4}\leavevmode {\color {Chapter }PositionNthOccurrence}}{287}{subsection.21.16.4}
\contentsline {subsection}{\numberline {21.16.5}\leavevmode {\color {Chapter }PositionSorted}}{287}{subsection.21.16.5}
\contentsline {subsection}{\numberline {21.16.6}\leavevmode {\color {Chapter }PositionSortedBy}}{288}{subsection.21.16.6}
\contentsline {subsection}{\numberline {21.16.7}\leavevmode {\color {Chapter }PositionSet}}{288}{subsection.21.16.7}
\contentsline {subsection}{\numberline {21.16.8}\leavevmode {\color {Chapter }PositionMaximum}}{289}{subsection.21.16.8}
\contentsline {subsection}{\numberline {21.16.9}\leavevmode {\color {Chapter }PositionProperty}}{289}{subsection.21.16.9}
\contentsline {subsection}{\numberline {21.16.10}\leavevmode {\color {Chapter }PositionsProperty}}{289}{subsection.21.16.10}
\contentsline {subsection}{\numberline {21.16.11}\leavevmode {\color {Chapter }PositionBound}}{290}{subsection.21.16.11}
\contentsline {subsection}{\numberline {21.16.12}\leavevmode {\color {Chapter }PositionsBound}}{290}{subsection.21.16.12}
\contentsline {subsection}{\numberline {21.16.13}\leavevmode {\color {Chapter }PositionNot}}{290}{subsection.21.16.13}
\contentsline {subsection}{\numberline {21.16.14}\leavevmode {\color {Chapter }PositionNonZero}}{290}{subsection.21.16.14}
\contentsline {subsection}{\numberline {21.16.15}\leavevmode {\color {Chapter }PositionSublist}}{291}{subsection.21.16.15}
\contentsline {section}{\numberline {21.17}\leavevmode {\color {Chapter }Properties and Attributes for Lists}}{291}{section.21.17}
\contentsline {subsection}{\numberline {21.17.1}\leavevmode {\color {Chapter }IsMatchingSublist}}{291}{subsection.21.17.1}
\contentsline {subsection}{\numberline {21.17.2}\leavevmode {\color {Chapter }IsDuplicateFree}}{292}{subsection.21.17.2}
\contentsline {subsection}{\numberline {21.17.3}\leavevmode {\color {Chapter }IsSortedList}}{292}{subsection.21.17.3}
\contentsline {subsection}{\numberline {21.17.4}\leavevmode {\color {Chapter }IsSSortedList}}{292}{subsection.21.17.4}
\contentsline {subsection}{\numberline {21.17.5}\leavevmode {\color {Chapter }Length}}{293}{subsection.21.17.5}
\contentsline {subsection}{\numberline {21.17.6}\leavevmode {\color {Chapter }ConstantTimeAccessList}}{293}{subsection.21.17.6}
\contentsline {section}{\numberline {21.18}\leavevmode {\color {Chapter }Sorting Lists}}{293}{section.21.18}
\contentsline {subsection}{\numberline {21.18.1}\leavevmode {\color {Chapter }Sort}}{293}{subsection.21.18.1}
\contentsline {subsection}{\numberline {21.18.2}\leavevmode {\color {Chapter }SortParallel}}{294}{subsection.21.18.2}
\contentsline {subsection}{\numberline {21.18.3}\leavevmode {\color {Chapter }Sortex}}{294}{subsection.21.18.3}
\contentsline {subsection}{\numberline {21.18.4}\leavevmode {\color {Chapter }SortingPerm}}{295}{subsection.21.18.4}
\contentsline {section}{\numberline {21.19}\leavevmode {\color {Chapter }Sorted Lists and Sets}}{295}{section.21.19}
\contentsline {subsection}{\numberline {21.19.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }in (for strictly sorted lists)}}{296}{subsection.21.19.1}
\contentsline {subsection}{\numberline {21.19.2}\leavevmode {\color {Chapter }IsEqualSet}}{296}{subsection.21.19.2}
\contentsline {subsection}{\numberline {21.19.3}\leavevmode {\color {Chapter }IsSubsetSet}}{296}{subsection.21.19.3}
\contentsline {subsection}{\numberline {21.19.4}\leavevmode {\color {Chapter }AddSet}}{297}{subsection.21.19.4}
\contentsline {subsection}{\numberline {21.19.5}\leavevmode {\color {Chapter }RemoveSet}}{297}{subsection.21.19.5}
\contentsline {subsection}{\numberline {21.19.6}\leavevmode {\color {Chapter }UniteSet}}{297}{subsection.21.19.6}
\contentsline {subsection}{\numberline {21.19.7}\leavevmode {\color {Chapter }IntersectSet}}{298}{subsection.21.19.7}
\contentsline {subsection}{\numberline {21.19.8}\leavevmode {\color {Chapter }SubtractSet}}{298}{subsection.21.19.8}
\contentsline {section}{\numberline {21.20}\leavevmode {\color {Chapter }Operations for Lists}}{298}{section.21.20}
\contentsline {subsection}{\numberline {21.20.1}\leavevmode {\color {Chapter }Concatenation (for several lists)}}{298}{subsection.21.20.1}
\contentsline {subsection}{\numberline {21.20.2}\leavevmode {\color {Chapter }Compacted}}{299}{subsection.21.20.2}
\contentsline {subsection}{\numberline {21.20.3}\leavevmode {\color {Chapter }Collected}}{299}{subsection.21.20.3}
\contentsline {subsection}{\numberline {21.20.4}\leavevmode {\color {Chapter }DuplicateFreeList}}{299}{subsection.21.20.4}
\contentsline {subsection}{\numberline {21.20.5}\leavevmode {\color {Chapter }AsDuplicateFreeList}}{300}{subsection.21.20.5}
\contentsline {subsection}{\numberline {21.20.6}\leavevmode {\color {Chapter }Flat}}{300}{subsection.21.20.6}
\contentsline {subsection}{\numberline {21.20.7}\leavevmode {\color {Chapter }Reversed}}{300}{subsection.21.20.7}
\contentsline {subsection}{\numberline {21.20.8}\leavevmode {\color {Chapter }Shuffle}}{301}{subsection.21.20.8}
\contentsline {subsection}{\numberline {21.20.9}\leavevmode {\color {Chapter }IsLexicographicallyLess}}{301}{subsection.21.20.9}
\contentsline {subsection}{\numberline {21.20.10}\leavevmode {\color {Chapter }Apply}}{301}{subsection.21.20.10}
\contentsline {subsection}{\numberline {21.20.11}\leavevmode {\color {Chapter }Perform}}{301}{subsection.21.20.11}
\contentsline {subsection}{\numberline {21.20.12}\leavevmode {\color {Chapter }PermListList}}{302}{subsection.21.20.12}
\contentsline {subsection}{\numberline {21.20.13}\leavevmode {\color {Chapter }Maximum}}{302}{subsection.21.20.13}
\contentsline {subsection}{\numberline {21.20.14}\leavevmode {\color {Chapter }Minimum}}{302}{subsection.21.20.14}
\contentsline {subsection}{\numberline {21.20.15}\leavevmode {\color {Chapter }MaximumList and MinimumList}}{303}{subsection.21.20.15}
\contentsline {subsection}{\numberline {21.20.16}\leavevmode {\color {Chapter }Cartesian}}{303}{subsection.21.20.16}
\contentsline {subsection}{\numberline {21.20.17}\leavevmode {\color {Chapter }IteratorOfCartesianProduct}}{304}{subsection.21.20.17}
\contentsline {subsection}{\numberline {21.20.18}\leavevmode {\color {Chapter }Permuted}}{304}{subsection.21.20.18}
\contentsline {subsection}{\numberline {21.20.19}\leavevmode {\color {Chapter }List (for a list (and a function))}}{304}{subsection.21.20.19}
\contentsline {subsection}{\numberline {21.20.20}\leavevmode {\color {Chapter }Filtered}}{305}{subsection.21.20.20}
\contentsline {subsection}{\numberline {21.20.21}\leavevmode {\color {Chapter }Number}}{305}{subsection.21.20.21}
\contentsline {subsection}{\numberline {21.20.22}\leavevmode {\color {Chapter }First}}{306}{subsection.21.20.22}
\contentsline {subsection}{\numberline {21.20.23}\leavevmode {\color {Chapter }ForAll}}{306}{subsection.21.20.23}
\contentsline {subsection}{\numberline {21.20.24}\leavevmode {\color {Chapter }ForAny}}{306}{subsection.21.20.24}
\contentsline {subsection}{\numberline {21.20.25}\leavevmode {\color {Chapter }Product}}{307}{subsection.21.20.25}
\contentsline {subsection}{\numberline {21.20.26}\leavevmode {\color {Chapter }Sum}}{307}{subsection.21.20.26}
\contentsline {subsection}{\numberline {21.20.27}\leavevmode {\color {Chapter }Iterated}}{308}{subsection.21.20.27}
\contentsline {subsection}{\numberline {21.20.28}\leavevmode {\color {Chapter }ListN}}{308}{subsection.21.20.28}
\contentsline {section}{\numberline {21.21}\leavevmode {\color {Chapter }Advanced List Manipulations}}{308}{section.21.21}
\contentsline {subsection}{\numberline {21.21.1}\leavevmode {\color {Chapter }ListX}}{308}{subsection.21.21.1}
\contentsline {subsection}{\numberline {21.21.2}\leavevmode {\color {Chapter }SetX}}{310}{subsection.21.21.2}
\contentsline {subsection}{\numberline {21.21.3}\leavevmode {\color {Chapter }SumX}}{310}{subsection.21.21.3}
\contentsline {subsection}{\numberline {21.21.4}\leavevmode {\color {Chapter }ProductX}}{310}{subsection.21.21.4}
\contentsline {section}{\numberline {21.22}\leavevmode {\color {Chapter }Ranges}}{310}{section.21.22}
\contentsline {subsection}{\numberline {21.22.1}\leavevmode {\color {Chapter }IsRange}}{311}{subsection.21.22.1}
\contentsline {subsection}{\numberline {21.22.2}\leavevmode {\color {Chapter }IsRangeRep}}{312}{subsection.21.22.2}
\contentsline {subsection}{\numberline {21.22.3}\leavevmode {\color {Chapter }ConvertToRangeRep}}{312}{subsection.21.22.3}
\contentsline {section}{\numberline {21.23}\leavevmode {\color {Chapter }Enumerators}}{313}{section.21.23}
\contentsline {subsection}{\numberline {21.23.1}\leavevmode {\color {Chapter }IsQuickPositionList}}{313}{subsection.21.23.1}
\contentsline {section}{\numberline {21.24}\leavevmode {\color {Chapter }Plain Lists}}{313}{section.21.24}
\contentsline {subsection}{\numberline {21.24.1}\leavevmode {\color {Chapter }IsPlistRep}}{313}{subsection.21.24.1}
\contentsline {chapter}{\numberline {22}\leavevmode {\color {Chapter }Boolean Lists}}{315}{chapter.22}
\contentsline {section}{\numberline {22.1}\leavevmode {\color {Chapter }IsBlist (Filter)}}{315}{section.22.1}
\contentsline {subsection}{\numberline {22.1.1}\leavevmode {\color {Chapter }IsBlist}}{315}{subsection.22.1.1}
\contentsline {section}{\numberline {22.2}\leavevmode {\color {Chapter }Boolean Lists Representing Subsets}}{316}{section.22.2}
\contentsline {subsection}{\numberline {22.2.1}\leavevmode {\color {Chapter }BlistList}}{316}{subsection.22.2.1}
\contentsline {subsection}{\numberline {22.2.2}\leavevmode {\color {Chapter }ListBlist}}{316}{subsection.22.2.2}
\contentsline {subsection}{\numberline {22.2.3}\leavevmode {\color {Chapter }SizeBlist}}{316}{subsection.22.2.3}
\contentsline {subsection}{\numberline {22.2.4}\leavevmode {\color {Chapter }IsSubsetBlist}}{317}{subsection.22.2.4}
\contentsline {section}{\numberline {22.3}\leavevmode {\color {Chapter }Set Operations via Boolean Lists}}{317}{section.22.3}
\contentsline {subsection}{\numberline {22.3.1}\leavevmode {\color {Chapter }UnionBlist}}{317}{subsection.22.3.1}
\contentsline {subsection}{\numberline {22.3.2}\leavevmode {\color {Chapter }IntersectionBlist}}{317}{subsection.22.3.2}
\contentsline {subsection}{\numberline {22.3.3}\leavevmode {\color {Chapter }DifferenceBlist}}{317}{subsection.22.3.3}
\contentsline {section}{\numberline {22.4}\leavevmode {\color {Chapter }Function that Modify Boolean Lists}}{318}{section.22.4}
\contentsline {subsection}{\numberline {22.4.1}\leavevmode {\color {Chapter }UniteBlist}}{318}{subsection.22.4.1}
\contentsline {subsection}{\numberline {22.4.2}\leavevmode {\color {Chapter }UniteBlistList}}{318}{subsection.22.4.2}
\contentsline {subsection}{\numberline {22.4.3}\leavevmode {\color {Chapter }IntersectBlist}}{318}{subsection.22.4.3}
\contentsline {subsection}{\numberline {22.4.4}\leavevmode {\color {Chapter }SubtractBlist}}{319}{subsection.22.4.4}
\contentsline {subsection}{\numberline {22.4.5}\leavevmode {\color {Chapter }FlipBlist}}{319}{subsection.22.4.5}
\contentsline {subsection}{\numberline {22.4.6}\leavevmode {\color {Chapter }SetAllBlist}}{319}{subsection.22.4.6}
\contentsline {subsection}{\numberline {22.4.7}\leavevmode {\color {Chapter }ClearAllBlist}}{320}{subsection.22.4.7}
\contentsline {section}{\numberline {22.5}\leavevmode {\color {Chapter }More about Boolean Lists}}{320}{section.22.5}
\contentsline {subsection}{\numberline {22.5.1}\leavevmode {\color {Chapter }IsBlistRep}}{321}{subsection.22.5.1}
\contentsline {chapter}{\numberline {23}\leavevmode {\color {Chapter }Row Vectors}}{322}{chapter.23}
\contentsline {section}{\numberline {23.1}\leavevmode {\color {Chapter }IsRowVector (Filter)}}{322}{section.23.1}
\contentsline {subsection}{\numberline {23.1.1}\leavevmode {\color {Chapter }IsRowVector}}{322}{subsection.23.1.1}
\contentsline {section}{\numberline {23.2}\leavevmode {\color {Chapter }Operators for Row Vectors}}{323}{section.23.2}
\contentsline {subsection}{\numberline {23.2.1}\leavevmode {\color {Chapter }NormedRowVector}}{324}{subsection.23.2.1}
\contentsline {section}{\numberline {23.3}\leavevmode {\color {Chapter }Row Vectors over Finite Fields}}{324}{section.23.3}
\contentsline {subsection}{\numberline {23.3.1}\leavevmode {\color {Chapter }ConvertToVectorRep}}{325}{subsection.23.3.1}
\contentsline {subsection}{\numberline {23.3.2}\leavevmode {\color {Chapter }ImmutableVector}}{326}{subsection.23.3.2}
\contentsline {subsection}{\numberline {23.3.3}\leavevmode {\color {Chapter }NumberFFVector}}{326}{subsection.23.3.3}
\contentsline {section}{\numberline {23.4}\leavevmode {\color {Chapter }Coefficient List Arithmetic}}{326}{section.23.4}
\contentsline {subsection}{\numberline {23.4.1}\leavevmode {\color {Chapter }AddRowVector}}{326}{subsection.23.4.1}
\contentsline {subsection}{\numberline {23.4.2}\leavevmode {\color {Chapter }AddCoeffs}}{327}{subsection.23.4.2}
\contentsline {subsection}{\numberline {23.4.3}\leavevmode {\color {Chapter }MultVector}}{327}{subsection.23.4.3}
\contentsline {subsection}{\numberline {23.4.4}\leavevmode {\color {Chapter }CoeffsMod}}{327}{subsection.23.4.4}
\contentsline {section}{\numberline {23.5}\leavevmode {\color {Chapter }Shifting and Trimming Coefficient Lists}}{327}{section.23.5}
\contentsline {subsection}{\numberline {23.5.1}\leavevmode {\color {Chapter }LeftShiftRowVector}}{327}{subsection.23.5.1}
\contentsline {subsection}{\numberline {23.5.2}\leavevmode {\color {Chapter }RightShiftRowVector}}{328}{subsection.23.5.2}
\contentsline {subsection}{\numberline {23.5.3}\leavevmode {\color {Chapter }ShrinkRowVector}}{328}{subsection.23.5.3}
\contentsline {subsection}{\numberline {23.5.4}\leavevmode {\color {Chapter }RemoveOuterCoeffs}}{328}{subsection.23.5.4}
\contentsline {section}{\numberline {23.6}\leavevmode {\color {Chapter }Functions for Coding Theory}}{328}{section.23.6}
\contentsline {subsection}{\numberline {23.6.1}\leavevmode {\color {Chapter }WeightVecFFE}}{328}{subsection.23.6.1}
\contentsline {subsection}{\numberline {23.6.2}\leavevmode {\color {Chapter }DistanceVecFFE}}{328}{subsection.23.6.2}
\contentsline {subsection}{\numberline {23.6.3}\leavevmode {\color {Chapter }DistancesDistributionVecFFEsVecFFE}}{329}{subsection.23.6.3}
\contentsline {subsection}{\numberline {23.6.4}\leavevmode {\color {Chapter }DistancesDistributionMatFFEVecFFE}}{329}{subsection.23.6.4}
\contentsline {subsection}{\numberline {23.6.5}\leavevmode {\color {Chapter }AClosestVectorCombinationsMatFFEVecFFE}}{329}{subsection.23.6.5}
\contentsline {subsection}{\numberline {23.6.6}\leavevmode {\color {Chapter }CosetLeadersMatFFE}}{329}{subsection.23.6.6}
\contentsline {section}{\numberline {23.7}\leavevmode {\color {Chapter }Vectors as coefficients of polynomials}}{330}{section.23.7}
\contentsline {subsection}{\numberline {23.7.1}\leavevmode {\color {Chapter }ValuePol}}{330}{subsection.23.7.1}
\contentsline {subsection}{\numberline {23.7.2}\leavevmode {\color {Chapter }ProductCoeffs}}{330}{subsection.23.7.2}
\contentsline {subsection}{\numberline {23.7.3}\leavevmode {\color {Chapter }ReduceCoeffs}}{330}{subsection.23.7.3}
\contentsline {subsection}{\numberline {23.7.4}\leavevmode {\color {Chapter }ReduceCoeffsMod}}{331}{subsection.23.7.4}
\contentsline {subsection}{\numberline {23.7.5}\leavevmode {\color {Chapter }PowerModCoeffs}}{331}{subsection.23.7.5}
\contentsline {subsection}{\numberline {23.7.6}\leavevmode {\color {Chapter }ShiftedCoeffs}}{331}{subsection.23.7.6}
\contentsline {chapter}{\numberline {24}\leavevmode {\color {Chapter }Matrices}}{332}{chapter.24}
\contentsline {section}{\numberline {24.1}\leavevmode {\color {Chapter }InfoMatrix (Info Class)}}{332}{section.24.1}
\contentsline {subsection}{\numberline {24.1.1}\leavevmode {\color {Chapter }InfoMatrix}}{332}{subsection.24.1.1}
\contentsline {section}{\numberline {24.2}\leavevmode {\color {Chapter }Categories of Matrices}}{332}{section.24.2}
\contentsline {subsection}{\numberline {24.2.1}\leavevmode {\color {Chapter }IsMatrix}}{332}{subsection.24.2.1}
\contentsline {subsection}{\numberline {24.2.2}\leavevmode {\color {Chapter }IsOrdinaryMatrix}}{333}{subsection.24.2.2}
\contentsline {subsection}{\numberline {24.2.3}\leavevmode {\color {Chapter }IsLieMatrix}}{333}{subsection.24.2.3}
\contentsline {section}{\numberline {24.3}\leavevmode {\color {Chapter }Operators for Matrices}}{334}{section.24.3}
\contentsline {section}{\numberline {24.4}\leavevmode {\color {Chapter }Properties and Attributes of Matrices}}{336}{section.24.4}
\contentsline {subsection}{\numberline {24.4.1}\leavevmode {\color {Chapter }DimensionsMat}}{336}{subsection.24.4.1}
\contentsline {subsection}{\numberline {24.4.2}\leavevmode {\color {Chapter }DefaultFieldOfMatrix}}{336}{subsection.24.4.2}
\contentsline {subsection}{\numberline {24.4.3}\leavevmode {\color {Chapter }TraceMat}}{336}{subsection.24.4.3}
\contentsline {subsection}{\numberline {24.4.4}\leavevmode {\color {Chapter }DeterminantMat}}{337}{subsection.24.4.4}
\contentsline {subsection}{\numberline {24.4.5}\leavevmode {\color {Chapter }DeterminantMatDestructive}}{337}{subsection.24.4.5}
\contentsline {subsection}{\numberline {24.4.6}\leavevmode {\color {Chapter }DeterminantMatDivFree}}{337}{subsection.24.4.6}
\contentsline {subsection}{\numberline {24.4.7}\leavevmode {\color {Chapter }IsEmptyMatrix (for matrices)}}{337}{subsection.24.4.7}
\contentsline {subsection}{\numberline {24.4.8}\leavevmode {\color {Chapter }IsMonomialMatrix}}{337}{subsection.24.4.8}
\contentsline {subsection}{\numberline {24.4.9}\leavevmode {\color {Chapter }IsDiagonalMatrix}}{338}{subsection.24.4.9}
\contentsline {subsection}{\numberline {24.4.10}\leavevmode {\color {Chapter }IsUpperTriangularMatrix}}{338}{subsection.24.4.10}
\contentsline {subsection}{\numberline {24.4.11}\leavevmode {\color {Chapter }IsLowerTriangularMatrix}}{338}{subsection.24.4.11}
\contentsline {section}{\numberline {24.5}\leavevmode {\color {Chapter }Matrix Constructions}}{338}{section.24.5}
\contentsline {subsection}{\numberline {24.5.1}\leavevmode {\color {Chapter }IdentityMat}}{338}{subsection.24.5.1}
\contentsline {subsection}{\numberline {24.5.2}\leavevmode {\color {Chapter }NullMat}}{339}{subsection.24.5.2}
\contentsline {subsection}{\numberline {24.5.3}\leavevmode {\color {Chapter }EmptyMatrix}}{339}{subsection.24.5.3}
\contentsline {subsection}{\numberline {24.5.4}\leavevmode {\color {Chapter }DiagonalMat}}{339}{subsection.24.5.4}
\contentsline {subsection}{\numberline {24.5.5}\leavevmode {\color {Chapter }PermutationMat}}{340}{subsection.24.5.5}
\contentsline {subsection}{\numberline {24.5.6}\leavevmode {\color {Chapter }TransposedMatImmutable}}{340}{subsection.24.5.6}
\contentsline {subsection}{\numberline {24.5.7}\leavevmode {\color {Chapter }TransposedMatDestructive}}{340}{subsection.24.5.7}
\contentsline {subsection}{\numberline {24.5.8}\leavevmode {\color {Chapter }KroneckerProduct}}{340}{subsection.24.5.8}
\contentsline {subsection}{\numberline {24.5.9}\leavevmode {\color {Chapter }ReflectionMat}}{341}{subsection.24.5.9}
\contentsline {subsection}{\numberline {24.5.10}\leavevmode {\color {Chapter }PrintArray}}{341}{subsection.24.5.10}
\contentsline {section}{\numberline {24.6}\leavevmode {\color {Chapter }Random Matrices}}{341}{section.24.6}
\contentsline {subsection}{\numberline {24.6.1}\leavevmode {\color {Chapter }RandomMat}}{341}{subsection.24.6.1}
\contentsline {subsection}{\numberline {24.6.2}\leavevmode {\color {Chapter }RandomInvertibleMat}}{341}{subsection.24.6.2}
\contentsline {subsection}{\numberline {24.6.3}\leavevmode {\color {Chapter }RandomUnimodularMat}}{342}{subsection.24.6.3}
\contentsline {section}{\numberline {24.7}\leavevmode {\color {Chapter }Matrices Representing Linear Equations and the Gaussian Algorithm}}{342}{section.24.7}
\contentsline {subsection}{\numberline {24.7.1}\leavevmode {\color {Chapter }RankMat}}{342}{subsection.24.7.1}
\contentsline {subsection}{\numberline {24.7.2}\leavevmode {\color {Chapter }TriangulizedMat}}{342}{subsection.24.7.2}
\contentsline {subsection}{\numberline {24.7.3}\leavevmode {\color {Chapter }TriangulizeMat}}{343}{subsection.24.7.3}
\contentsline {subsection}{\numberline {24.7.4}\leavevmode {\color {Chapter }NullspaceMat}}{343}{subsection.24.7.4}
\contentsline {subsection}{\numberline {24.7.5}\leavevmode {\color {Chapter }NullspaceMatDestructive}}{343}{subsection.24.7.5}
\contentsline {subsection}{\numberline {24.7.6}\leavevmode {\color {Chapter }SolutionMat}}{344}{subsection.24.7.6}
\contentsline {subsection}{\numberline {24.7.7}\leavevmode {\color {Chapter }SolutionMatDestructive}}{344}{subsection.24.7.7}
\contentsline {subsection}{\numberline {24.7.8}\leavevmode {\color {Chapter }BaseFixedSpace}}{344}{subsection.24.7.8}
\contentsline {section}{\numberline {24.8}\leavevmode {\color {Chapter }Eigenvectors and eigenvalues}}{344}{section.24.8}
\contentsline {subsection}{\numberline {24.8.1}\leavevmode {\color {Chapter }GeneralisedEigenvalues}}{344}{subsection.24.8.1}
\contentsline {subsection}{\numberline {24.8.2}\leavevmode {\color {Chapter }GeneralisedEigenspaces}}{345}{subsection.24.8.2}
\contentsline {subsection}{\numberline {24.8.3}\leavevmode {\color {Chapter }Eigenvalues}}{345}{subsection.24.8.3}
\contentsline {subsection}{\numberline {24.8.4}\leavevmode {\color {Chapter }Eigenspaces}}{345}{subsection.24.8.4}
\contentsline {subsection}{\numberline {24.8.5}\leavevmode {\color {Chapter }Eigenvectors}}{345}{subsection.24.8.5}
\contentsline {section}{\numberline {24.9}\leavevmode {\color {Chapter }Elementary Divisors}}{345}{section.24.9}
\contentsline {subsection}{\numberline {24.9.1}\leavevmode {\color {Chapter }ElementaryDivisorsMat}}{345}{subsection.24.9.1}
\contentsline {subsection}{\numberline {24.9.2}\leavevmode {\color {Chapter }ElementaryDivisorsTransformationsMat}}{346}{subsection.24.9.2}
\contentsline {subsection}{\numberline {24.9.3}\leavevmode {\color {Chapter }DiagonalizeMat}}{347}{subsection.24.9.3}
\contentsline {section}{\numberline {24.10}\leavevmode {\color {Chapter }Echelonized Matrices}}{347}{section.24.10}
\contentsline {subsection}{\numberline {24.10.1}\leavevmode {\color {Chapter }SemiEchelonMat}}{347}{subsection.24.10.1}
\contentsline {subsection}{\numberline {24.10.2}\leavevmode {\color {Chapter }SemiEchelonMatDestructive}}{347}{subsection.24.10.2}
\contentsline {subsection}{\numberline {24.10.3}\leavevmode {\color {Chapter }SemiEchelonMatTransformation}}{348}{subsection.24.10.3}
\contentsline {subsection}{\numberline {24.10.4}\leavevmode {\color {Chapter }SemiEchelonMats}}{348}{subsection.24.10.4}
\contentsline {subsection}{\numberline {24.10.5}\leavevmode {\color {Chapter }SemiEchelonMatsDestructive}}{348}{subsection.24.10.5}
\contentsline {section}{\numberline {24.11}\leavevmode {\color {Chapter }Matrices as Basis of a Row Space}}{349}{section.24.11}
\contentsline {subsection}{\numberline {24.11.1}\leavevmode {\color {Chapter }BaseMat}}{349}{subsection.24.11.1}
\contentsline {subsection}{\numberline {24.11.2}\leavevmode {\color {Chapter }BaseMatDestructive}}{349}{subsection.24.11.2}
\contentsline {subsection}{\numberline {24.11.3}\leavevmode {\color {Chapter }BaseOrthogonalSpaceMat}}{349}{subsection.24.11.3}
\contentsline {subsection}{\numberline {24.11.4}\leavevmode {\color {Chapter }SumIntersectionMat}}{349}{subsection.24.11.4}
\contentsline {subsection}{\numberline {24.11.5}\leavevmode {\color {Chapter }BaseSteinitzVectors}}{350}{subsection.24.11.5}
\contentsline {section}{\numberline {24.12}\leavevmode {\color {Chapter }Triangular Matrices}}{350}{section.24.12}
\contentsline {subsection}{\numberline {24.12.1}\leavevmode {\color {Chapter }DiagonalOfMatrix}}{350}{subsection.24.12.1}
\contentsline {subsection}{\numberline {24.12.2}\leavevmode {\color {Chapter }UpperSubdiagonal}}{350}{subsection.24.12.2}
\contentsline {subsection}{\numberline {24.12.3}\leavevmode {\color {Chapter }DepthOfUpperTriangularMatrix}}{351}{subsection.24.12.3}
\contentsline {section}{\numberline {24.13}\leavevmode {\color {Chapter }Matrices as Linear Mappings}}{351}{section.24.13}
\contentsline {subsection}{\numberline {24.13.1}\leavevmode {\color {Chapter }CharacteristicPolynomial}}{351}{subsection.24.13.1}
\contentsline {subsection}{\numberline {24.13.2}\leavevmode {\color {Chapter }RationalCanonicalFormTransform}}{352}{subsection.24.13.2}
\contentsline {subsection}{\numberline {24.13.3}\leavevmode {\color {Chapter }JordanDecomposition}}{352}{subsection.24.13.3}
\contentsline {subsection}{\numberline {24.13.4}\leavevmode {\color {Chapter }BlownUpMat}}{352}{subsection.24.13.4}
\contentsline {subsection}{\numberline {24.13.5}\leavevmode {\color {Chapter }BlownUpVector}}{353}{subsection.24.13.5}
\contentsline {subsection}{\numberline {24.13.6}\leavevmode {\color {Chapter }CompanionMat}}{353}{subsection.24.13.6}
\contentsline {section}{\numberline {24.14}\leavevmode {\color {Chapter }Matrices over Finite Fields}}{353}{section.24.14}
\contentsline {subsection}{\numberline {24.14.1}\leavevmode {\color {Chapter }ImmutableMatrix}}{355}{subsection.24.14.1}
\contentsline {subsection}{\numberline {24.14.2}\leavevmode {\color {Chapter }ConvertToMatrixRep (for a list (and a field))}}{355}{subsection.24.14.2}
\contentsline {subsection}{\numberline {24.14.3}\leavevmode {\color {Chapter }ProjectiveOrder}}{356}{subsection.24.14.3}
\contentsline {subsection}{\numberline {24.14.4}\leavevmode {\color {Chapter }SimultaneousEigenvalues}}{356}{subsection.24.14.4}
\contentsline {section}{\numberline {24.15}\leavevmode {\color {Chapter }Inverse and Nullspace of an Integer Matrix Modulo an Ideal}}{356}{section.24.15}
\contentsline {subsection}{\numberline {24.15.1}\leavevmode {\color {Chapter }InverseMatMod}}{356}{subsection.24.15.1}
\contentsline {subsection}{\numberline {24.15.2}\leavevmode {\color {Chapter }BasisNullspaceModN}}{356}{subsection.24.15.2}
\contentsline {subsection}{\numberline {24.15.3}\leavevmode {\color {Chapter }NullspaceModQ}}{357}{subsection.24.15.3}
\contentsline {section}{\numberline {24.16}\leavevmode {\color {Chapter }Special Multiplication Algorithms for Matrices over GF(2)}}{357}{section.24.16}
\contentsline {subsection}{\numberline {24.16.1}\leavevmode {\color {Chapter }PROD{\textunderscore }GF2MAT{\textunderscore }GF2MAT{\textunderscore }SIMPLE}}{357}{subsection.24.16.1}
\contentsline {subsection}{\numberline {24.16.2}\leavevmode {\color {Chapter }PROD{\textunderscore }GF2MAT{\textunderscore }GF2MAT{\textunderscore }ADVANCED}}{357}{subsection.24.16.2}
\contentsline {section}{\numberline {24.17}\leavevmode {\color {Chapter }Block Matrices}}{358}{section.24.17}
\contentsline {subsection}{\numberline {24.17.1}\leavevmode {\color {Chapter }AsBlockMatrix}}{358}{subsection.24.17.1}
\contentsline {subsection}{\numberline {24.17.2}\leavevmode {\color {Chapter }BlockMatrix}}{358}{subsection.24.17.2}
\contentsline {subsection}{\numberline {24.17.3}\leavevmode {\color {Chapter }MatrixByBlockMatrix}}{358}{subsection.24.17.3}
\contentsline {section}{\numberline {24.18}\leavevmode {\color {Chapter }Linear Programming}}{358}{section.24.18}
\contentsline {subsection}{\numberline {24.18.1}\leavevmode {\color {Chapter }SimplexMethod}}{358}{subsection.24.18.1}
\contentsline {chapter}{\numberline {25}\leavevmode {\color {Chapter }Integral matrices and lattices}}{360}{chapter.25}
\contentsline {section}{\numberline {25.1}\leavevmode {\color {Chapter }Linear equations over the integers and Integral Matrices}}{360}{section.25.1}
\contentsline {subsection}{\numberline {25.1.1}\leavevmode {\color {Chapter }NullspaceIntMat}}{360}{subsection.25.1.1}
\contentsline {subsection}{\numberline {25.1.2}\leavevmode {\color {Chapter }SolutionIntMat}}{360}{subsection.25.1.2}
\contentsline {subsection}{\numberline {25.1.3}\leavevmode {\color {Chapter }SolutionNullspaceIntMat}}{360}{subsection.25.1.3}
\contentsline {subsection}{\numberline {25.1.4}\leavevmode {\color {Chapter }BaseIntMat}}{361}{subsection.25.1.4}
\contentsline {subsection}{\numberline {25.1.5}\leavevmode {\color {Chapter }BaseIntersectionIntMats}}{361}{subsection.25.1.5}
\contentsline {subsection}{\numberline {25.1.6}\leavevmode {\color {Chapter }ComplementIntMat}}{361}{subsection.25.1.6}
\contentsline {section}{\numberline {25.2}\leavevmode {\color {Chapter }Normal Forms over the Integers}}{362}{section.25.2}
\contentsline {subsection}{\numberline {25.2.1}\leavevmode {\color {Chapter }TriangulizedIntegerMat}}{362}{subsection.25.2.1}
\contentsline {subsection}{\numberline {25.2.2}\leavevmode {\color {Chapter }TriangulizedIntegerMatTransform}}{362}{subsection.25.2.2}
\contentsline {subsection}{\numberline {25.2.3}\leavevmode {\color {Chapter }TriangulizeIntegerMat}}{362}{subsection.25.2.3}
\contentsline {subsection}{\numberline {25.2.4}\leavevmode {\color {Chapter }HermiteNormalFormIntegerMat}}{363}{subsection.25.2.4}
\contentsline {subsection}{\numberline {25.2.5}\leavevmode {\color {Chapter }HermiteNormalFormIntegerMatTransform}}{363}{subsection.25.2.5}
\contentsline {subsection}{\numberline {25.2.6}\leavevmode {\color {Chapter }SmithNormalFormIntegerMat}}{363}{subsection.25.2.6}
\contentsline {subsection}{\numberline {25.2.7}\leavevmode {\color {Chapter }SmithNormalFormIntegerMatTransforms}}{363}{subsection.25.2.7}
\contentsline {subsection}{\numberline {25.2.8}\leavevmode {\color {Chapter }DiagonalizeIntMat}}{364}{subsection.25.2.8}
\contentsline {subsection}{\numberline {25.2.9}\leavevmode {\color {Chapter }NormalFormIntMat}}{364}{subsection.25.2.9}
\contentsline {subsection}{\numberline {25.2.10}\leavevmode {\color {Chapter }AbelianInvariantsOfList}}{365}{subsection.25.2.10}
\contentsline {section}{\numberline {25.3}\leavevmode {\color {Chapter }Determinant of an integer matrix}}{365}{section.25.3}
\contentsline {subsection}{\numberline {25.3.1}\leavevmode {\color {Chapter }DeterminantIntMat}}{365}{subsection.25.3.1}
\contentsline {section}{\numberline {25.4}\leavevmode {\color {Chapter }Decompositions}}{365}{section.25.4}
\contentsline {subsection}{\numberline {25.4.1}\leavevmode {\color {Chapter }Decomposition}}{366}{subsection.25.4.1}
\contentsline {subsection}{\numberline {25.4.2}\leavevmode {\color {Chapter }LinearIndependentColumns}}{366}{subsection.25.4.2}
\contentsline {subsection}{\numberline {25.4.3}\leavevmode {\color {Chapter }PadicCoefficients}}{367}{subsection.25.4.3}
\contentsline {subsection}{\numberline {25.4.4}\leavevmode {\color {Chapter }IntegralizedMat}}{367}{subsection.25.4.4}
\contentsline {subsection}{\numberline {25.4.5}\leavevmode {\color {Chapter }DecompositionInt}}{367}{subsection.25.4.5}
\contentsline {section}{\numberline {25.5}\leavevmode {\color {Chapter }Lattice Reduction}}{367}{section.25.5}
\contentsline {subsection}{\numberline {25.5.1}\leavevmode {\color {Chapter }LLLReducedBasis}}{367}{subsection.25.5.1}
\contentsline {subsection}{\numberline {25.5.2}\leavevmode {\color {Chapter }LLLReducedGramMat}}{368}{subsection.25.5.2}
\contentsline {section}{\numberline {25.6}\leavevmode {\color {Chapter }Orthogonal Embeddings}}{369}{section.25.6}
\contentsline {subsection}{\numberline {25.6.1}\leavevmode {\color {Chapter }OrthogonalEmbeddings}}{369}{subsection.25.6.1}
\contentsline {subsection}{\numberline {25.6.2}\leavevmode {\color {Chapter }ShortestVectors}}{369}{subsection.25.6.2}
\contentsline {chapter}{\numberline {26}\leavevmode {\color {Chapter }Vector and Matrix Objects}}{371}{chapter.26}
\contentsline {section}{\numberline {26.1}\leavevmode {\color {Chapter }Fundamental Ideas and Rules}}{371}{section.26.1}
\contentsline {section}{\numberline {26.2}\leavevmode {\color {Chapter }Categories of Vectors and Matrices}}{372}{section.26.2}
\contentsline {section}{\numberline {26.3}\leavevmode {\color {Chapter }Constructing Vector and Matrix Objects}}{372}{section.26.3}
\contentsline {section}{\numberline {26.4}\leavevmode {\color {Chapter }Operations for Vector Objects}}{372}{section.26.4}
\contentsline {subsection}{\numberline {26.4.1}\leavevmode {\color {Chapter }PositionNonZero (for vectors)}}{372}{subsection.26.4.1}
\contentsline {subsection}{\numberline {26.4.2}\leavevmode {\color {Chapter }PositionLastNonZero}}{372}{subsection.26.4.2}
\contentsline {subsection}{\numberline {26.4.3}\leavevmode {\color {Chapter }ListOp (for IsVectorObj, IsFunction)}}{372}{subsection.26.4.3}
\contentsline {subsection}{\numberline {26.4.4}\leavevmode {\color {Chapter }Unpack (for IsVectorObj)}}{373}{subsection.26.4.4}
\contentsline {subsection}{\numberline {26.4.5}\leavevmode {\color {Chapter }ConcatenationOfVectors (for IsVectorObj)}}{373}{subsection.26.4.5}
\contentsline {subsection}{\numberline {26.4.6}\leavevmode {\color {Chapter }ExtractSubVector (for IsVectorObj,IsList)}}{373}{subsection.26.4.6}
\contentsline {subsection}{\numberline {26.4.7}\leavevmode {\color {Chapter }ZeroVector (for IsInt,IsVectorObj)}}{373}{subsection.26.4.7}
\contentsline {subsection}{\numberline {26.4.8}\leavevmode {\color {Chapter }ConstructingFilter (for IsVectorObj)}}{373}{subsection.26.4.8}
\contentsline {subsection}{\numberline {26.4.9}\leavevmode {\color {Chapter }Randomize}}{373}{subsection.26.4.9}
\contentsline {subsection}{\numberline {26.4.10}\leavevmode {\color {Chapter }WeightOfVector (for IsVectorObj)}}{373}{subsection.26.4.10}
\contentsline {subsection}{\numberline {26.4.11}\leavevmode {\color {Chapter }DistanceOfVectors (for IsVectorObj,IsVectorObj)}}{374}{subsection.26.4.11}
\contentsline {section}{\numberline {26.5}\leavevmode {\color {Chapter }Operations for Row List Matrix Objects}}{374}{section.26.5}
\contentsline {section}{\numberline {26.6}\leavevmode {\color {Chapter }Operations for Flat Matrix Objects}}{374}{section.26.6}
\contentsline {chapter}{\numberline {27}\leavevmode {\color {Chapter }Strings and Characters}}{375}{chapter.27}
\contentsline {section}{\numberline {27.1}\leavevmode {\color {Chapter }IsChar and IsString}}{375}{section.27.1}
\contentsline {subsection}{\numberline {27.1.1}\leavevmode {\color {Chapter }IsChar}}{375}{subsection.27.1.1}
\contentsline {subsection}{\numberline {27.1.2}\leavevmode {\color {Chapter }IsString}}{375}{subsection.27.1.2}
\contentsline {subsection}{\numberline {27.1.3}\leavevmode {\color {Chapter }Strings As Lists}}{376}{subsection.27.1.3}
\contentsline {subsection}{\numberline {27.1.4}\leavevmode {\color {Chapter }Printing Strings}}{376}{subsection.27.1.4}
\contentsline {section}{\numberline {27.2}\leavevmode {\color {Chapter }Special Characters}}{377}{section.27.2}
\contentsline {section}{\numberline {27.3}\leavevmode {\color {Chapter }Triple Quoted Strings}}{379}{section.27.3}
\contentsline {section}{\numberline {27.4}\leavevmode {\color {Chapter }Internally Represented Strings}}{379}{section.27.4}
\contentsline {subsection}{\numberline {27.4.1}\leavevmode {\color {Chapter }IsStringRep}}{379}{subsection.27.4.1}
\contentsline {subsection}{\numberline {27.4.2}\leavevmode {\color {Chapter }ConvertToStringRep}}{379}{subsection.27.4.2}
\contentsline {subsection}{\numberline {27.4.3}\leavevmode {\color {Chapter }CopyToStringRep}}{379}{subsection.27.4.3}
\contentsline {subsection}{\numberline {27.4.4}\leavevmode {\color {Chapter }IsEmptyString}}{380}{subsection.27.4.4}
\contentsline {subsection}{\numberline {27.4.5}\leavevmode {\color {Chapter }EmptyString}}{380}{subsection.27.4.5}
\contentsline {subsection}{\numberline {27.4.6}\leavevmode {\color {Chapter }CharsFamily}}{380}{subsection.27.4.6}
\contentsline {section}{\numberline {27.5}\leavevmode {\color {Chapter }Recognizing Characters}}{381}{section.27.5}
\contentsline {subsection}{\numberline {27.5.1}\leavevmode {\color {Chapter }IsDigitChar}}{381}{subsection.27.5.1}
\contentsline {subsection}{\numberline {27.5.2}\leavevmode {\color {Chapter }IsLowerAlphaChar}}{381}{subsection.27.5.2}
\contentsline {subsection}{\numberline {27.5.3}\leavevmode {\color {Chapter }IsUpperAlphaChar}}{381}{subsection.27.5.3}
\contentsline {subsection}{\numberline {27.5.4}\leavevmode {\color {Chapter }IsAlphaChar}}{381}{subsection.27.5.4}
\contentsline {section}{\numberline {27.6}\leavevmode {\color {Chapter }Comparisons of Strings}}{381}{section.27.6}
\contentsline {subsection}{\numberline {27.6.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= (for two strings)}}{381}{subsection.27.6.1}
\contentsline {subsection}{\numberline {27.6.2}\leavevmode {\color {Chapter }\texttt {\char 92\relax }{\textless } (for two strings)}}{382}{subsection.27.6.2}
\contentsline {section}{\numberline {27.7}\leavevmode {\color {Chapter }Operations to Produce or Manipulate Strings}}{382}{section.27.7}
\contentsline {subsection}{\numberline {27.7.1}\leavevmode {\color {Chapter }DisplayString}}{382}{subsection.27.7.1}
\contentsline {subsection}{\numberline {27.7.2}\leavevmode {\color {Chapter }DEFAULTDISPLAYSTRING}}{382}{subsection.27.7.2}
\contentsline {subsection}{\numberline {27.7.3}\leavevmode {\color {Chapter }ViewString}}{383}{subsection.27.7.3}
\contentsline {subsection}{\numberline {27.7.4}\leavevmode {\color {Chapter }DEFAULTVIEWSTRING}}{383}{subsection.27.7.4}
\contentsline {subsection}{\numberline {27.7.5}\leavevmode {\color {Chapter }PrintString}}{383}{subsection.27.7.5}
\contentsline {subsection}{\numberline {27.7.6}\leavevmode {\color {Chapter }String}}{383}{subsection.27.7.6}
\contentsline {subsection}{\numberline {27.7.7}\leavevmode {\color {Chapter }StripLineBreakCharacters}}{384}{subsection.27.7.7}
\contentsline {subsection}{\numberline {27.7.8}\leavevmode {\color {Chapter }HexStringInt}}{384}{subsection.27.7.8}
\contentsline {subsection}{\numberline {27.7.9}\leavevmode {\color {Chapter }StringPP}}{384}{subsection.27.7.9}
\contentsline {subsection}{\numberline {27.7.10}\leavevmode {\color {Chapter }WordAlp}}{384}{subsection.27.7.10}
\contentsline {subsection}{\numberline {27.7.11}\leavevmode {\color {Chapter }LowercaseString}}{384}{subsection.27.7.11}
\contentsline {subsection}{\numberline {27.7.12}\leavevmode {\color {Chapter }LowercaseChar}}{385}{subsection.27.7.12}
\contentsline {subsection}{\numberline {27.7.13}\leavevmode {\color {Chapter }UppercaseString}}{385}{subsection.27.7.13}
\contentsline {subsection}{\numberline {27.7.14}\leavevmode {\color {Chapter }UppercaseChar}}{385}{subsection.27.7.14}
\contentsline {subsection}{\numberline {27.7.15}\leavevmode {\color {Chapter }SplitString}}{385}{subsection.27.7.15}
\contentsline {subsection}{\numberline {27.7.16}\leavevmode {\color {Chapter }ReplacedString}}{386}{subsection.27.7.16}
\contentsline {subsection}{\numberline {27.7.17}\leavevmode {\color {Chapter }NormalizeWhitespace}}{386}{subsection.27.7.17}
\contentsline {subsection}{\numberline {27.7.18}\leavevmode {\color {Chapter }NormalizedWhitespace}}{386}{subsection.27.7.18}
\contentsline {subsection}{\numberline {27.7.19}\leavevmode {\color {Chapter }RemoveCharacters}}{386}{subsection.27.7.19}
\contentsline {subsection}{\numberline {27.7.20}\leavevmode {\color {Chapter }JoinStringsWithSeparator}}{387}{subsection.27.7.20}
\contentsline {subsection}{\numberline {27.7.21}\leavevmode {\color {Chapter }Chomp}}{387}{subsection.27.7.21}
\contentsline {subsection}{\numberline {27.7.22}\leavevmode {\color {Chapter }StartsWith}}{388}{subsection.27.7.22}
\contentsline {subsection}{\numberline {27.7.23}\leavevmode {\color {Chapter }StringFormatted}}{388}{subsection.27.7.23}
\contentsline {subsection}{\numberline {27.7.24}\leavevmode {\color {Chapter }NumbersString}}{389}{subsection.27.7.24}
\contentsline {subsection}{\numberline {27.7.25}\leavevmode {\color {Chapter }StringNumbers}}{389}{subsection.27.7.25}
\contentsline {subsection}{\numberline {27.7.26}\leavevmode {\color {Chapter }StringOfMemoryAmount}}{389}{subsection.27.7.26}
\contentsline {section}{\numberline {27.8}\leavevmode {\color {Chapter }Character Conversion}}{390}{section.27.8}
\contentsline {subsection}{\numberline {27.8.1}\leavevmode {\color {Chapter }IntChar}}{390}{subsection.27.8.1}
\contentsline {subsection}{\numberline {27.8.2}\leavevmode {\color {Chapter }CharInt}}{390}{subsection.27.8.2}
\contentsline {subsection}{\numberline {27.8.3}\leavevmode {\color {Chapter }SIntChar}}{390}{subsection.27.8.3}
\contentsline {subsection}{\numberline {27.8.4}\leavevmode {\color {Chapter }CharSInt}}{390}{subsection.27.8.4}
\contentsline {section}{\numberline {27.9}\leavevmode {\color {Chapter }Operations to Evaluate Strings}}{391}{section.27.9}
\contentsline {subsection}{\numberline {27.9.1}\leavevmode {\color {Chapter }Int (for strings)}}{391}{subsection.27.9.1}
\contentsline {subsection}{\numberline {27.9.2}\leavevmode {\color {Chapter }Rat (for strings)}}{391}{subsection.27.9.2}
\contentsline {subsection}{\numberline {27.9.3}\leavevmode {\color {Chapter }IntHexString}}{391}{subsection.27.9.3}
\contentsline {subsection}{\numberline {27.9.4}\leavevmode {\color {Chapter }Ordinal}}{392}{subsection.27.9.4}
\contentsline {subsection}{\numberline {27.9.5}\leavevmode {\color {Chapter }EvalString}}{392}{subsection.27.9.5}
\contentsline {subsection}{\numberline {27.9.6}\leavevmode {\color {Chapter }CrcString}}{393}{subsection.27.9.6}
\contentsline {section}{\numberline {27.10}\leavevmode {\color {Chapter }Calendar Arithmetic}}{393}{section.27.10}
\contentsline {subsection}{\numberline {27.10.1}\leavevmode {\color {Chapter }DaysInYear}}{393}{subsection.27.10.1}
\contentsline {subsection}{\numberline {27.10.2}\leavevmode {\color {Chapter }DaysInMonth}}{393}{subsection.27.10.2}
\contentsline {subsection}{\numberline {27.10.3}\leavevmode {\color {Chapter }DMYDay}}{393}{subsection.27.10.3}
\contentsline {subsection}{\numberline {27.10.4}\leavevmode {\color {Chapter }DayDMY}}{393}{subsection.27.10.4}
\contentsline {subsection}{\numberline {27.10.5}\leavevmode {\color {Chapter }WeekDay}}{394}{subsection.27.10.5}
\contentsline {subsection}{\numberline {27.10.6}\leavevmode {\color {Chapter }StringDate}}{394}{subsection.27.10.6}
\contentsline {subsection}{\numberline {27.10.7}\leavevmode {\color {Chapter }HMSMSec}}{394}{subsection.27.10.7}
\contentsline {subsection}{\numberline {27.10.8}\leavevmode {\color {Chapter }SecHMSM}}{394}{subsection.27.10.8}
\contentsline {subsection}{\numberline {27.10.9}\leavevmode {\color {Chapter }StringTime}}{394}{subsection.27.10.9}
\contentsline {subsection}{\numberline {27.10.10}\leavevmode {\color {Chapter }SecondsDMYhms}}{395}{subsection.27.10.10}
\contentsline {subsection}{\numberline {27.10.11}\leavevmode {\color {Chapter }DMYhmsSeconds}}{395}{subsection.27.10.11}
\contentsline {section}{\numberline {27.11}\leavevmode {\color {Chapter }Obtaining LaTeX Representations of Objects}}{395}{section.27.11}
\contentsline {chapter}{\numberline {28}\leavevmode {\color {Chapter }Dictionaries and General Hash Tables}}{396}{chapter.28}
\contentsline {section}{\numberline {28.1}\leavevmode {\color {Chapter }Using Dictionaries}}{396}{section.28.1}
\contentsline {section}{\numberline {28.2}\leavevmode {\color {Chapter }Dictionaries}}{398}{section.28.2}
\contentsline {subsection}{\numberline {28.2.1}\leavevmode {\color {Chapter }NewDictionary}}{398}{subsection.28.2.1}
\contentsline {section}{\numberline {28.3}\leavevmode {\color {Chapter }Dictionaries via Binary Lists}}{398}{section.28.3}
\contentsline {subsection}{\numberline {28.3.1}\leavevmode {\color {Chapter }DictionaryByPosition}}{398}{subsection.28.3.1}
\contentsline {subsection}{\numberline {28.3.2}\leavevmode {\color {Chapter }IsDictionary}}{399}{subsection.28.3.2}
\contentsline {subsection}{\numberline {28.3.3}\leavevmode {\color {Chapter }IsLookupDictionary}}{399}{subsection.28.3.3}
\contentsline {subsection}{\numberline {28.3.4}\leavevmode {\color {Chapter }AddDictionary}}{399}{subsection.28.3.4}
\contentsline {subsection}{\numberline {28.3.5}\leavevmode {\color {Chapter }KnowsDictionary}}{399}{subsection.28.3.5}
\contentsline {subsection}{\numberline {28.3.6}\leavevmode {\color {Chapter }LookupDictionary}}{399}{subsection.28.3.6}
\contentsline {section}{\numberline {28.4}\leavevmode {\color {Chapter }General Hash Tables}}{399}{section.28.4}
\contentsline {section}{\numberline {28.5}\leavevmode {\color {Chapter }Hash keys}}{400}{section.28.5}
\contentsline {subsection}{\numberline {28.5.1}\leavevmode {\color {Chapter }DenseIntKey}}{400}{subsection.28.5.1}
\contentsline {subsection}{\numberline {28.5.2}\leavevmode {\color {Chapter }SparseIntKey}}{400}{subsection.28.5.2}
\contentsline {section}{\numberline {28.6}\leavevmode {\color {Chapter }Dense hash tables}}{400}{section.28.6}
\contentsline {subsection}{\numberline {28.6.1}\leavevmode {\color {Chapter }DenseHashTable}}{400}{subsection.28.6.1}
\contentsline {section}{\numberline {28.7}\leavevmode {\color {Chapter }Sparse hash tables}}{400}{section.28.7}
\contentsline {subsection}{\numberline {28.7.1}\leavevmode {\color {Chapter }SparseHashTable}}{401}{subsection.28.7.1}
\contentsline {subsection}{\numberline {28.7.2}\leavevmode {\color {Chapter }DoubleHashArraySize}}{401}{subsection.28.7.2}
\contentsline {chapter}{\numberline {29}\leavevmode {\color {Chapter }Records}}{402}{chapter.29}
\contentsline {section}{\numberline {29.1}\leavevmode {\color {Chapter }IsRecord and RecNames}}{402}{section.29.1}
\contentsline {subsection}{\numberline {29.1.1}\leavevmode {\color {Chapter }IsRecord}}{402}{subsection.29.1.1}
\contentsline {subsection}{\numberline {29.1.2}\leavevmode {\color {Chapter }RecNames}}{403}{subsection.29.1.2}
\contentsline {section}{\numberline {29.2}\leavevmode {\color {Chapter }Accessing Record Elements}}{403}{section.29.2}
\contentsline {section}{\numberline {29.3}\leavevmode {\color {Chapter }Record Assignment}}{404}{section.29.3}
\contentsline {section}{\numberline {29.4}\leavevmode {\color {Chapter }Identical Records}}{404}{section.29.4}
\contentsline {section}{\numberline {29.5}\leavevmode {\color {Chapter }Comparisons of Records}}{406}{section.29.5}
\contentsline {section}{\numberline {29.6}\leavevmode {\color {Chapter }IsBound and Unbind for Records}}{407}{section.29.6}
\contentsline {subsection}{\numberline {29.6.1}\leavevmode {\color {Chapter }IsBound (for a record component)}}{407}{subsection.29.6.1}
\contentsline {subsection}{\numberline {29.6.2}\leavevmode {\color {Chapter }Unbind (unbind a record component)}}{407}{subsection.29.6.2}
\contentsline {section}{\numberline {29.7}\leavevmode {\color {Chapter }Record Access Operations}}{408}{section.29.7}
\contentsline {subsection}{\numberline {29.7.1}\leavevmode {\color {Chapter }NameRNam}}{408}{subsection.29.7.1}
\contentsline {subsection}{\numberline {29.7.2}\leavevmode {\color {Chapter }RNamObj (for a string)}}{408}{subsection.29.7.2}
\contentsline {subsection}{\numberline {29.7.3}\leavevmode {\color {Chapter }\texttt {\char 92\relax }.}}{408}{subsection.29.7.3}
\contentsline {chapter}{\numberline {30}\leavevmode {\color {Chapter }Collections}}{410}{chapter.30}
\contentsline {section}{\numberline {30.1}\leavevmode {\color {Chapter }IsCollection (Filter)}}{410}{section.30.1}
\contentsline {subsection}{\numberline {30.1.1}\leavevmode {\color {Chapter }IsCollection}}{410}{subsection.30.1.1}
\contentsline {section}{\numberline {30.2}\leavevmode {\color {Chapter }Collection Families}}{410}{section.30.2}
\contentsline {subsection}{\numberline {30.2.1}\leavevmode {\color {Chapter }CollectionsFamily}}{410}{subsection.30.2.1}
\contentsline {subsection}{\numberline {30.2.2}\leavevmode {\color {Chapter }IsCollectionFamily}}{411}{subsection.30.2.2}
\contentsline {subsection}{\numberline {30.2.3}\leavevmode {\color {Chapter }ElementsFamily}}{411}{subsection.30.2.3}
\contentsline {subsection}{\numberline {30.2.4}\leavevmode {\color {Chapter }CategoryCollections}}{411}{subsection.30.2.4}
\contentsline {subsection}{\numberline {30.2.5}\leavevmode {\color {Chapter }DeclareCategoryCollections}}{411}{subsection.30.2.5}
\contentsline {section}{\numberline {30.3}\leavevmode {\color {Chapter }Lists and Collections}}{412}{section.30.3}
\contentsline {subsection}{\numberline {30.3.1}\leavevmode {\color {Chapter }IsListOrCollection}}{412}{subsection.30.3.1}
\contentsline {subsection}{\numberline {30.3.2}\leavevmode {\color {Chapter }Enumerator}}{412}{subsection.30.3.2}
\contentsline {subsection}{\numberline {30.3.3}\leavevmode {\color {Chapter }EnumeratorSorted}}{412}{subsection.30.3.3}
\contentsline {subsection}{\numberline {30.3.4}\leavevmode {\color {Chapter }EnumeratorByFunctions (for a domain and a record)}}{413}{subsection.30.3.4}
\contentsline {subsection}{\numberline {30.3.5}\leavevmode {\color {Chapter }List (for a collection)}}{414}{subsection.30.3.5}
\contentsline {subsection}{\numberline {30.3.6}\leavevmode {\color {Chapter }SortedList}}{415}{subsection.30.3.6}
\contentsline {subsection}{\numberline {30.3.7}\leavevmode {\color {Chapter }SSortedList}}{415}{subsection.30.3.7}
\contentsline {subsection}{\numberline {30.3.8}\leavevmode {\color {Chapter }AsList}}{416}{subsection.30.3.8}
\contentsline {subsection}{\numberline {30.3.9}\leavevmode {\color {Chapter }AsSortedList}}{416}{subsection.30.3.9}
\contentsline {subsection}{\numberline {30.3.10}\leavevmode {\color {Chapter }AsSSortedList}}{417}{subsection.30.3.10}
\contentsline {subsection}{\numberline {30.3.11}\leavevmode {\color {Chapter }Elements}}{417}{subsection.30.3.11}
\contentsline {section}{\numberline {30.4}\leavevmode {\color {Chapter }Attributes and Properties for Collections}}{418}{section.30.4}
\contentsline {subsection}{\numberline {30.4.1}\leavevmode {\color {Chapter }IsEmpty}}{418}{subsection.30.4.1}
\contentsline {subsection}{\numberline {30.4.2}\leavevmode {\color {Chapter }IsFinite}}{418}{subsection.30.4.2}
\contentsline {subsection}{\numberline {30.4.3}\leavevmode {\color {Chapter }IsTrivial}}{418}{subsection.30.4.3}
\contentsline {subsection}{\numberline {30.4.4}\leavevmode {\color {Chapter }IsNonTrivial}}{418}{subsection.30.4.4}
\contentsline {subsection}{\numberline {30.4.5}\leavevmode {\color {Chapter }IsWholeFamily}}{418}{subsection.30.4.5}
\contentsline {subsection}{\numberline {30.4.6}\leavevmode {\color {Chapter }Size}}{419}{subsection.30.4.6}
\contentsline {subsection}{\numberline {30.4.7}\leavevmode {\color {Chapter }Representative}}{419}{subsection.30.4.7}
\contentsline {subsection}{\numberline {30.4.8}\leavevmode {\color {Chapter }RepresentativeSmallest}}{419}{subsection.30.4.8}
\contentsline {section}{\numberline {30.5}\leavevmode {\color {Chapter }Operations for Collections}}{420}{section.30.5}
\contentsline {subsection}{\numberline {30.5.1}\leavevmode {\color {Chapter }IsSubset}}{420}{subsection.30.5.1}
\contentsline {subsection}{\numberline {30.5.2}\leavevmode {\color {Chapter }Intersection}}{420}{subsection.30.5.2}
\contentsline {subsection}{\numberline {30.5.3}\leavevmode {\color {Chapter }Union}}{421}{subsection.30.5.3}
\contentsline {subsection}{\numberline {30.5.4}\leavevmode {\color {Chapter }Difference}}{422}{subsection.30.5.4}
\contentsline {section}{\numberline {30.6}\leavevmode {\color {Chapter }Membership Test for Collections}}{422}{section.30.6}
\contentsline {subsection}{\numberline {30.6.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }in (for a collection)}}{422}{subsection.30.6.1}
\contentsline {section}{\numberline {30.7}\leavevmode {\color {Chapter }Random Elements}}{422}{section.30.7}
\contentsline {subsection}{\numberline {30.7.1}\leavevmode {\color {Chapter }Random (for a list or collection)}}{422}{subsection.30.7.1}
\contentsline {subsection}{\numberline {30.7.2}\leavevmode {\color {Chapter }PseudoRandom}}{423}{subsection.30.7.2}
\contentsline {subsection}{\numberline {30.7.3}\leavevmode {\color {Chapter }RandomList}}{423}{subsection.30.7.3}
\contentsline {section}{\numberline {30.8}\leavevmode {\color {Chapter }Iterators}}{424}{section.30.8}
\contentsline {subsection}{\numberline {30.8.1}\leavevmode {\color {Chapter }Iterator}}{424}{subsection.30.8.1}
\contentsline {subsection}{\numberline {30.8.2}\leavevmode {\color {Chapter }IteratorSorted}}{425}{subsection.30.8.2}
\contentsline {subsection}{\numberline {30.8.3}\leavevmode {\color {Chapter }IsIterator}}{425}{subsection.30.8.3}
\contentsline {subsection}{\numberline {30.8.4}\leavevmode {\color {Chapter }IsDoneIterator}}{425}{subsection.30.8.4}
\contentsline {subsection}{\numberline {30.8.5}\leavevmode {\color {Chapter }NextIterator}}{425}{subsection.30.8.5}
\contentsline {subsection}{\numberline {30.8.6}\leavevmode {\color {Chapter }IteratorList}}{426}{subsection.30.8.6}
\contentsline {subsection}{\numberline {30.8.7}\leavevmode {\color {Chapter }TrivialIterator}}{426}{subsection.30.8.7}
\contentsline {subsection}{\numberline {30.8.8}\leavevmode {\color {Chapter }IteratorByFunctions}}{426}{subsection.30.8.8}
\contentsline {chapter}{\numberline {31}\leavevmode {\color {Chapter }Domains and their Elements}}{428}{chapter.31}
\contentsline {section}{\numberline {31.1}\leavevmode {\color {Chapter }Operational Structure of Domains}}{428}{section.31.1}
\contentsline {section}{\numberline {31.2}\leavevmode {\color {Chapter }Equality and Comparison of Domains}}{429}{section.31.2}
\contentsline {section}{\numberline {31.3}\leavevmode {\color {Chapter }Constructing Domains}}{430}{section.31.3}
\contentsline {section}{\numberline {31.4}\leavevmode {\color {Chapter }Changing the Structure}}{431}{section.31.4}
\contentsline {section}{\numberline {31.5}\leavevmode {\color {Chapter }Changing the Representation}}{431}{section.31.5}
\contentsline {section}{\numberline {31.6}\leavevmode {\color {Chapter }Domain Categories}}{432}{section.31.6}
\contentsline {section}{\numberline {31.7}\leavevmode {\color {Chapter }Parents}}{433}{section.31.7}
\contentsline {subsection}{\numberline {31.7.1}\leavevmode {\color {Chapter }Parent}}{433}{subsection.31.7.1}
\contentsline {section}{\numberline {31.8}\leavevmode {\color {Chapter }Constructing Subdomains}}{433}{section.31.8}
\contentsline {section}{\numberline {31.9}\leavevmode {\color {Chapter }Operations for Domains}}{434}{section.31.9}
\contentsline {subsection}{\numberline {31.9.1}\leavevmode {\color {Chapter }IsGeneralizedDomain}}{434}{subsection.31.9.1}
\contentsline {subsection}{\numberline {31.9.2}\leavevmode {\color {Chapter }GeneratorsOfDomain}}{434}{subsection.31.9.2}
\contentsline {subsection}{\numberline {31.9.3}\leavevmode {\color {Chapter }Domain}}{435}{subsection.31.9.3}
\contentsline {section}{\numberline {31.10}\leavevmode {\color {Chapter }Attributes and Properties of Elements}}{435}{section.31.10}
\contentsline {subsection}{\numberline {31.10.1}\leavevmode {\color {Chapter }Characteristic}}{435}{subsection.31.10.1}
\contentsline {subsection}{\numberline {31.10.2}\leavevmode {\color {Chapter }OneImmutable}}{436}{subsection.31.10.2}
\contentsline {subsection}{\numberline {31.10.3}\leavevmode {\color {Chapter }ZeroImmutable}}{436}{subsection.31.10.3}
\contentsline {subsection}{\numberline {31.10.4}\leavevmode {\color {Chapter }MultiplicativeZeroOp}}{437}{subsection.31.10.4}
\contentsline {subsection}{\numberline {31.10.5}\leavevmode {\color {Chapter }IsOne}}{438}{subsection.31.10.5}
\contentsline {subsection}{\numberline {31.10.6}\leavevmode {\color {Chapter }IsZero}}{438}{subsection.31.10.6}
\contentsline {subsection}{\numberline {31.10.7}\leavevmode {\color {Chapter }IsIdempotent}}{438}{subsection.31.10.7}
\contentsline {subsection}{\numberline {31.10.8}\leavevmode {\color {Chapter }InverseImmutable}}{438}{subsection.31.10.8}
\contentsline {subsection}{\numberline {31.10.9}\leavevmode {\color {Chapter }AdditiveInverseImmutable}}{439}{subsection.31.10.9}
\contentsline {subsection}{\numberline {31.10.10}\leavevmode {\color {Chapter }Order}}{439}{subsection.31.10.10}
\contentsline {section}{\numberline {31.11}\leavevmode {\color {Chapter }Comparison Operations for Elements}}{439}{section.31.11}
\contentsline {subsection}{\numberline {31.11.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= and \texttt {\char 92\relax }{\textless }}}{439}{subsection.31.11.1}
\contentsline {subsection}{\numberline {31.11.2}\leavevmode {\color {Chapter }CanEasilyCompareElements}}{440}{subsection.31.11.2}
\contentsline {section}{\numberline {31.12}\leavevmode {\color {Chapter }Arithmetic Operations for Elements}}{440}{section.31.12}
\contentsline {subsection}{\numberline {31.12.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }+, \texttt {\char 92\relax }*, \texttt {\char 92\relax }/, \texttt {\char 92\relax }\texttt {\char 94\relax }, \texttt {\char 92\relax }mod}}{440}{subsection.31.12.1}
\contentsline {subsection}{\numberline {31.12.2}\leavevmode {\color {Chapter }LeftQuotient}}{441}{subsection.31.12.2}
\contentsline {subsection}{\numberline {31.12.3}\leavevmode {\color {Chapter }Comm}}{441}{subsection.31.12.3}
\contentsline {subsection}{\numberline {31.12.4}\leavevmode {\color {Chapter }LieBracket}}{441}{subsection.31.12.4}
\contentsline {subsection}{\numberline {31.12.5}\leavevmode {\color {Chapter }Sqrt}}{441}{subsection.31.12.5}
\contentsline {section}{\numberline {31.13}\leavevmode {\color {Chapter }Relations Between Domains}}{441}{section.31.13}
\contentsline {subsection}{\numberline {31.13.1}\leavevmode {\color {Chapter }UseSubsetRelation}}{442}{subsection.31.13.1}
\contentsline {subsection}{\numberline {31.13.2}\leavevmode {\color {Chapter }UseFactorRelation}}{442}{subsection.31.13.2}
\contentsline {subsection}{\numberline {31.13.3}\leavevmode {\color {Chapter }UseIsomorphismRelation}}{443}{subsection.31.13.3}
\contentsline {subsection}{\numberline {31.13.4}\leavevmode {\color {Chapter }InstallSubsetMaintenance}}{443}{subsection.31.13.4}
\contentsline {subsection}{\numberline {31.13.5}\leavevmode {\color {Chapter }InstallFactorMaintenance}}{444}{subsection.31.13.5}
\contentsline {subsection}{\numberline {31.13.6}\leavevmode {\color {Chapter }InstallIsomorphismMaintenance}}{444}{subsection.31.13.6}
\contentsline {section}{\numberline {31.14}\leavevmode {\color {Chapter }Useful Categories of Elements}}{444}{section.31.14}
\contentsline {subsection}{\numberline {31.14.1}\leavevmode {\color {Chapter }IsExtAElement}}{445}{subsection.31.14.1}
\contentsline {subsection}{\numberline {31.14.2}\leavevmode {\color {Chapter }IsNearAdditiveElement}}{445}{subsection.31.14.2}
\contentsline {subsection}{\numberline {31.14.3}\leavevmode {\color {Chapter }IsAdditiveElement}}{445}{subsection.31.14.3}
\contentsline {subsection}{\numberline {31.14.4}\leavevmode {\color {Chapter }IsNearAdditiveElementWithZero}}{445}{subsection.31.14.4}
\contentsline {subsection}{\numberline {31.14.5}\leavevmode {\color {Chapter }IsAdditiveElementWithZero}}{445}{subsection.31.14.5}
\contentsline {subsection}{\numberline {31.14.6}\leavevmode {\color {Chapter }IsNearAdditiveElementWithInverse}}{445}{subsection.31.14.6}
\contentsline {subsection}{\numberline {31.14.7}\leavevmode {\color {Chapter }IsAdditiveElementWithInverse}}{446}{subsection.31.14.7}
\contentsline {subsection}{\numberline {31.14.8}\leavevmode {\color {Chapter }IsExtLElement}}{446}{subsection.31.14.8}
\contentsline {subsection}{\numberline {31.14.9}\leavevmode {\color {Chapter }IsExtRElement}}{446}{subsection.31.14.9}
\contentsline {subsection}{\numberline {31.14.10}\leavevmode {\color {Chapter }IsMultiplicativeElement}}{446}{subsection.31.14.10}
\contentsline {subsection}{\numberline {31.14.11}\leavevmode {\color {Chapter }IsMultiplicativeElementWithOne}}{446}{subsection.31.14.11}
\contentsline {subsection}{\numberline {31.14.12}\leavevmode {\color {Chapter }IsMultiplicativeElementWithZero}}{446}{subsection.31.14.12}
\contentsline {subsection}{\numberline {31.14.13}\leavevmode {\color {Chapter }IsMultiplicativeElementWithInverse}}{447}{subsection.31.14.13}
\contentsline {subsection}{\numberline {31.14.14}\leavevmode {\color {Chapter }IsVector}}{447}{subsection.31.14.14}
\contentsline {subsection}{\numberline {31.14.15}\leavevmode {\color {Chapter }IsNearRingElement}}{447}{subsection.31.14.15}
\contentsline {subsection}{\numberline {31.14.16}\leavevmode {\color {Chapter }IsRingElement}}{447}{subsection.31.14.16}
\contentsline {subsection}{\numberline {31.14.17}\leavevmode {\color {Chapter }IsNearRingElementWithOne}}{447}{subsection.31.14.17}
\contentsline {subsection}{\numberline {31.14.18}\leavevmode {\color {Chapter }IsRingElementWithOne}}{448}{subsection.31.14.18}
\contentsline {subsection}{\numberline {31.14.19}\leavevmode {\color {Chapter }IsNearRingElementWithInverse}}{448}{subsection.31.14.19}
\contentsline {subsection}{\numberline {31.14.20}\leavevmode {\color {Chapter }IsRingElementWithInverse}}{448}{subsection.31.14.20}
\contentsline {section}{\numberline {31.15}\leavevmode {\color {Chapter }Useful Categories for all Elements of a Family}}{448}{section.31.15}
\contentsline {subsection}{\numberline {31.15.1}\leavevmode {\color {Chapter }IsAssociativeElement}}{448}{subsection.31.15.1}
\contentsline {subsection}{\numberline {31.15.2}\leavevmode {\color {Chapter }IsAdditivelyCommutativeElement}}{449}{subsection.31.15.2}
\contentsline {subsection}{\numberline {31.15.3}\leavevmode {\color {Chapter }IsCommutativeElement}}{449}{subsection.31.15.3}
\contentsline {subsection}{\numberline {31.15.4}\leavevmode {\color {Chapter }IsFiniteOrderElement}}{449}{subsection.31.15.4}
\contentsline {subsection}{\numberline {31.15.5}\leavevmode {\color {Chapter }IsJacobianElement}}{449}{subsection.31.15.5}
\contentsline {subsection}{\numberline {31.15.6}\leavevmode {\color {Chapter }IsZeroSquaredElement}}{450}{subsection.31.15.6}
\contentsline {chapter}{\numberline {32}\leavevmode {\color {Chapter }Mappings}}{451}{chapter.32}
\contentsline {section}{\numberline {32.1}\leavevmode {\color {Chapter }Direct Products and their Elements}}{451}{section.32.1}
\contentsline {subsection}{\numberline {32.1.1}\leavevmode {\color {Chapter }IsDirectProductElement}}{451}{subsection.32.1.1}
\contentsline {subsection}{\numberline {32.1.2}\leavevmode {\color {Chapter }DirectProductFamily}}{452}{subsection.32.1.2}
\contentsline {section}{\numberline {32.2}\leavevmode {\color {Chapter }Creating Mappings}}{453}{section.32.2}
\contentsline {subsection}{\numberline {32.2.1}\leavevmode {\color {Chapter }GeneralMappingByElements}}{453}{subsection.32.2.1}
\contentsline {subsection}{\numberline {32.2.2}\leavevmode {\color {Chapter }MappingByFunction}}{453}{subsection.32.2.2}
\contentsline {subsection}{\numberline {32.2.3}\leavevmode {\color {Chapter }InverseGeneralMapping}}{453}{subsection.32.2.3}
\contentsline {subsection}{\numberline {32.2.4}\leavevmode {\color {Chapter }RestrictedInverseGeneralMapping}}{453}{subsection.32.2.4}
\contentsline {subsection}{\numberline {32.2.5}\leavevmode {\color {Chapter }CompositionMapping}}{454}{subsection.32.2.5}
\contentsline {subsection}{\numberline {32.2.6}\leavevmode {\color {Chapter }CompositionMapping2}}{454}{subsection.32.2.6}
\contentsline {subsection}{\numberline {32.2.7}\leavevmode {\color {Chapter }IsCompositionMappingRep}}{454}{subsection.32.2.7}
\contentsline {subsection}{\numberline {32.2.8}\leavevmode {\color {Chapter }ConstituentsCompositionMapping}}{455}{subsection.32.2.8}
\contentsline {subsection}{\numberline {32.2.9}\leavevmode {\color {Chapter }ZeroMapping}}{455}{subsection.32.2.9}
\contentsline {subsection}{\numberline {32.2.10}\leavevmode {\color {Chapter }IdentityMapping}}{455}{subsection.32.2.10}
\contentsline {subsection}{\numberline {32.2.11}\leavevmode {\color {Chapter }Embedding}}{455}{subsection.32.2.11}
\contentsline {subsection}{\numberline {32.2.12}\leavevmode {\color {Chapter }Projection}}{455}{subsection.32.2.12}
\contentsline {subsection}{\numberline {32.2.13}\leavevmode {\color {Chapter }RestrictedMapping}}{456}{subsection.32.2.13}
\contentsline {section}{\numberline {32.3}\leavevmode {\color {Chapter }Properties and Attributes of (General) Mappings}}{456}{section.32.3}
\contentsline {subsection}{\numberline {32.3.1}\leavevmode {\color {Chapter }IsTotal}}{456}{subsection.32.3.1}
\contentsline {subsection}{\numberline {32.3.2}\leavevmode {\color {Chapter }IsSingleValued}}{456}{subsection.32.3.2}
\contentsline {subsection}{\numberline {32.3.3}\leavevmode {\color {Chapter }IsMapping}}{456}{subsection.32.3.3}
\contentsline {subsection}{\numberline {32.3.4}\leavevmode {\color {Chapter }IsInjective}}{456}{subsection.32.3.4}
\contentsline {subsection}{\numberline {32.3.5}\leavevmode {\color {Chapter }IsSurjective}}{456}{subsection.32.3.5}
\contentsline {subsection}{\numberline {32.3.6}\leavevmode {\color {Chapter }IsBijective}}{457}{subsection.32.3.6}
\contentsline {subsection}{\numberline {32.3.7}\leavevmode {\color {Chapter }Range (of a general mapping)}}{457}{subsection.32.3.7}
\contentsline {subsection}{\numberline {32.3.8}\leavevmode {\color {Chapter }Source}}{457}{subsection.32.3.8}
\contentsline {subsection}{\numberline {32.3.9}\leavevmode {\color {Chapter }UnderlyingRelation}}{457}{subsection.32.3.9}
\contentsline {subsection}{\numberline {32.3.10}\leavevmode {\color {Chapter }UnderlyingGeneralMapping}}{457}{subsection.32.3.10}
\contentsline {section}{\numberline {32.4}\leavevmode {\color {Chapter }Images under Mappings}}{457}{section.32.4}
\contentsline {subsection}{\numberline {32.4.1}\leavevmode {\color {Chapter }ImagesSource}}{457}{subsection.32.4.1}
\contentsline {subsection}{\numberline {32.4.2}\leavevmode {\color {Chapter }ImagesRepresentative}}{458}{subsection.32.4.2}
\contentsline {subsection}{\numberline {32.4.3}\leavevmode {\color {Chapter }ImagesElm}}{458}{subsection.32.4.3}
\contentsline {subsection}{\numberline {32.4.4}\leavevmode {\color {Chapter }ImagesSet}}{458}{subsection.32.4.4}
\contentsline {subsection}{\numberline {32.4.5}\leavevmode {\color {Chapter }ImageElm}}{458}{subsection.32.4.5}
\contentsline {subsection}{\numberline {32.4.6}\leavevmode {\color {Chapter }Image}}{458}{subsection.32.4.6}
\contentsline {subsection}{\numberline {32.4.7}\leavevmode {\color {Chapter }Images}}{459}{subsection.32.4.7}
\contentsline {section}{\numberline {32.5}\leavevmode {\color {Chapter }Preimages under Mappings}}{459}{section.32.5}
\contentsline {subsection}{\numberline {32.5.1}\leavevmode {\color {Chapter }PreImagesRange}}{459}{subsection.32.5.1}
\contentsline {subsection}{\numberline {32.5.2}\leavevmode {\color {Chapter }PreImagesElm}}{459}{subsection.32.5.2}
\contentsline {subsection}{\numberline {32.5.3}\leavevmode {\color {Chapter }PreImageElm}}{460}{subsection.32.5.3}
\contentsline {subsection}{\numberline {32.5.4}\leavevmode {\color {Chapter }PreImagesRepresentative}}{460}{subsection.32.5.4}
\contentsline {subsection}{\numberline {32.5.5}\leavevmode {\color {Chapter }PreImagesSet}}{460}{subsection.32.5.5}
\contentsline {subsection}{\numberline {32.5.6}\leavevmode {\color {Chapter }PreImage}}{460}{subsection.32.5.6}
\contentsline {subsection}{\numberline {32.5.7}\leavevmode {\color {Chapter }PreImages}}{461}{subsection.32.5.7}
\contentsline {section}{\numberline {32.6}\leavevmode {\color {Chapter }Arithmetic Operations for General Mappings}}{461}{section.32.6}
\contentsline {section}{\numberline {32.7}\leavevmode {\color {Chapter }Mappings which are Compatible with Algebraic Structures}}{462}{section.32.7}
\contentsline {section}{\numberline {32.8}\leavevmode {\color {Chapter }Magma Homomorphisms}}{462}{section.32.8}
\contentsline {subsection}{\numberline {32.8.1}\leavevmode {\color {Chapter }IsMagmaHomomorphism}}{462}{subsection.32.8.1}
\contentsline {subsection}{\numberline {32.8.2}\leavevmode {\color {Chapter }MagmaHomomorphismByFunctionNC}}{462}{subsection.32.8.2}
\contentsline {subsection}{\numberline {32.8.3}\leavevmode {\color {Chapter }NaturalHomomorphismByGenerators}}{462}{subsection.32.8.3}
\contentsline {section}{\numberline {32.9}\leavevmode {\color {Chapter }Mappings that Respect Multiplication}}{462}{section.32.9}
\contentsline {subsection}{\numberline {32.9.1}\leavevmode {\color {Chapter }RespectsMultiplication}}{462}{subsection.32.9.1}
\contentsline {subsection}{\numberline {32.9.2}\leavevmode {\color {Chapter }RespectsOne}}{463}{subsection.32.9.2}
\contentsline {subsection}{\numberline {32.9.3}\leavevmode {\color {Chapter }RespectsInverses}}{463}{subsection.32.9.3}
\contentsline {subsection}{\numberline {32.9.4}\leavevmode {\color {Chapter }IsGroupGeneralMapping}}{463}{subsection.32.9.4}
\contentsline {subsection}{\numberline {32.9.5}\leavevmode {\color {Chapter }KernelOfMultiplicativeGeneralMapping}}{463}{subsection.32.9.5}
\contentsline {subsection}{\numberline {32.9.6}\leavevmode {\color {Chapter }CoKernelOfMultiplicativeGeneralMapping}}{463}{subsection.32.9.6}
\contentsline {section}{\numberline {32.10}\leavevmode {\color {Chapter }Mappings that Respect Addition}}{464}{section.32.10}
\contentsline {subsection}{\numberline {32.10.1}\leavevmode {\color {Chapter }RespectsAddition}}{464}{subsection.32.10.1}
\contentsline {subsection}{\numberline {32.10.2}\leavevmode {\color {Chapter }RespectsAdditiveInverses}}{464}{subsection.32.10.2}
\contentsline {subsection}{\numberline {32.10.3}\leavevmode {\color {Chapter }RespectsZero}}{464}{subsection.32.10.3}
\contentsline {subsection}{\numberline {32.10.4}\leavevmode {\color {Chapter }IsAdditiveGroupGeneralMapping}}{464}{subsection.32.10.4}
\contentsline {subsection}{\numberline {32.10.5}\leavevmode {\color {Chapter }KernelOfAdditiveGeneralMapping}}{465}{subsection.32.10.5}
\contentsline {subsection}{\numberline {32.10.6}\leavevmode {\color {Chapter }CoKernelOfAdditiveGeneralMapping}}{465}{subsection.32.10.6}
\contentsline {section}{\numberline {32.11}\leavevmode {\color {Chapter }Linear Mappings}}{465}{section.32.11}
\contentsline {subsection}{\numberline {32.11.1}\leavevmode {\color {Chapter }RespectsScalarMultiplication}}{465}{subsection.32.11.1}
\contentsline {subsection}{\numberline {32.11.2}\leavevmode {\color {Chapter }IsLeftModuleGeneralMapping}}{465}{subsection.32.11.2}
\contentsline {subsection}{\numberline {32.11.3}\leavevmode {\color {Chapter }IsLinearMapping}}{465}{subsection.32.11.3}
\contentsline {section}{\numberline {32.12}\leavevmode {\color {Chapter }Ring Homomorphisms}}{466}{section.32.12}
\contentsline {subsection}{\numberline {32.12.1}\leavevmode {\color {Chapter }IsRingGeneralMapping}}{466}{subsection.32.12.1}
\contentsline {subsection}{\numberline {32.12.2}\leavevmode {\color {Chapter }IsRingWithOneGeneralMapping}}{466}{subsection.32.12.2}
\contentsline {subsection}{\numberline {32.12.3}\leavevmode {\color {Chapter }IsAlgebraGeneralMapping}}{466}{subsection.32.12.3}
\contentsline {subsection}{\numberline {32.12.4}\leavevmode {\color {Chapter }IsAlgebraWithOneGeneralMapping}}{466}{subsection.32.12.4}
\contentsline {subsection}{\numberline {32.12.5}\leavevmode {\color {Chapter }IsFieldHomomorphism}}{466}{subsection.32.12.5}
\contentsline {section}{\numberline {32.13}\leavevmode {\color {Chapter }General Mappings}}{467}{section.32.13}
\contentsline {subsection}{\numberline {32.13.1}\leavevmode {\color {Chapter }IsGeneralMapping}}{467}{subsection.32.13.1}
\contentsline {subsection}{\numberline {32.13.2}\leavevmode {\color {Chapter }IsConstantTimeAccessGeneralMapping}}{467}{subsection.32.13.2}
\contentsline {subsection}{\numberline {32.13.3}\leavevmode {\color {Chapter }IsEndoGeneralMapping}}{467}{subsection.32.13.3}
\contentsline {section}{\numberline {32.14}\leavevmode {\color {Chapter }Technical Matters Concerning General Mappings}}{467}{section.32.14}
\contentsline {subsection}{\numberline {32.14.1}\leavevmode {\color {Chapter }IsSPGeneralMapping}}{468}{subsection.32.14.1}
\contentsline {subsection}{\numberline {32.14.2}\leavevmode {\color {Chapter }IsGeneralMappingFamily}}{468}{subsection.32.14.2}
\contentsline {subsection}{\numberline {32.14.3}\leavevmode {\color {Chapter }FamilyRange}}{468}{subsection.32.14.3}
\contentsline {subsection}{\numberline {32.14.4}\leavevmode {\color {Chapter }FamilySource}}{469}{subsection.32.14.4}
\contentsline {subsection}{\numberline {32.14.5}\leavevmode {\color {Chapter }FamiliesOfGeneralMappingsAndRanges}}{469}{subsection.32.14.5}
\contentsline {subsection}{\numberline {32.14.6}\leavevmode {\color {Chapter }GeneralMappingsFamily}}{469}{subsection.32.14.6}
\contentsline {subsection}{\numberline {32.14.7}\leavevmode {\color {Chapter }TypeOfDefaultGeneralMapping}}{469}{subsection.32.14.7}
\contentsline {chapter}{\numberline {33}\leavevmode {\color {Chapter }Relations}}{470}{chapter.33}
\contentsline {section}{\numberline {33.1}\leavevmode {\color {Chapter }General Binary Relations}}{470}{section.33.1}
\contentsline {subsection}{\numberline {33.1.1}\leavevmode {\color {Chapter }IsBinaryRelation}}{470}{subsection.33.1.1}
\contentsline {subsection}{\numberline {33.1.2}\leavevmode {\color {Chapter }BinaryRelationByElements}}{470}{subsection.33.1.2}
\contentsline {subsection}{\numberline {33.1.3}\leavevmode {\color {Chapter }IdentityBinaryRelation}}{470}{subsection.33.1.3}
\contentsline {subsection}{\numberline {33.1.4}\leavevmode {\color {Chapter }EmptyBinaryRelation (for a degree)}}{471}{subsection.33.1.4}
\contentsline {section}{\numberline {33.2}\leavevmode {\color {Chapter }Properties and Attributes of Binary Relations}}{471}{section.33.2}
\contentsline {subsection}{\numberline {33.2.1}\leavevmode {\color {Chapter }IsReflexiveBinaryRelation}}{471}{subsection.33.2.1}
\contentsline {subsection}{\numberline {33.2.2}\leavevmode {\color {Chapter }IsSymmetricBinaryRelation}}{471}{subsection.33.2.2}
\contentsline {subsection}{\numberline {33.2.3}\leavevmode {\color {Chapter }IsTransitiveBinaryRelation}}{472}{subsection.33.2.3}
\contentsline {subsection}{\numberline {33.2.4}\leavevmode {\color {Chapter }IsAntisymmetricBinaryRelation}}{472}{subsection.33.2.4}
\contentsline {subsection}{\numberline {33.2.5}\leavevmode {\color {Chapter }IsPreOrderBinaryRelation}}{472}{subsection.33.2.5}
\contentsline {subsection}{\numberline {33.2.6}\leavevmode {\color {Chapter }IsPartialOrderBinaryRelation}}{472}{subsection.33.2.6}
\contentsline {subsection}{\numberline {33.2.7}\leavevmode {\color {Chapter }IsHasseDiagram}}{472}{subsection.33.2.7}
\contentsline {subsection}{\numberline {33.2.8}\leavevmode {\color {Chapter }IsEquivalenceRelation}}{473}{subsection.33.2.8}
\contentsline {subsection}{\numberline {33.2.9}\leavevmode {\color {Chapter }Successors}}{473}{subsection.33.2.9}
\contentsline {subsection}{\numberline {33.2.10}\leavevmode {\color {Chapter }DegreeOfBinaryRelation}}{473}{subsection.33.2.10}
\contentsline {subsection}{\numberline {33.2.11}\leavevmode {\color {Chapter }PartialOrderOfHasseDiagram}}{473}{subsection.33.2.11}
\contentsline {section}{\numberline {33.3}\leavevmode {\color {Chapter }Binary Relations on Points}}{473}{section.33.3}
\contentsline {subsection}{\numberline {33.3.1}\leavevmode {\color {Chapter }BinaryRelationOnPoints}}{474}{subsection.33.3.1}
\contentsline {subsection}{\numberline {33.3.2}\leavevmode {\color {Chapter }RandomBinaryRelationOnPoints}}{474}{subsection.33.3.2}
\contentsline {subsection}{\numberline {33.3.3}\leavevmode {\color {Chapter }AsBinaryRelationOnPoints}}{474}{subsection.33.3.3}
\contentsline {section}{\numberline {33.4}\leavevmode {\color {Chapter }Closure Operations and Other Constructors}}{474}{section.33.4}
\contentsline {subsection}{\numberline {33.4.1}\leavevmode {\color {Chapter }ReflexiveClosureBinaryRelation}}{474}{subsection.33.4.1}
\contentsline {subsection}{\numberline {33.4.2}\leavevmode {\color {Chapter }SymmetricClosureBinaryRelation}}{475}{subsection.33.4.2}
\contentsline {subsection}{\numberline {33.4.3}\leavevmode {\color {Chapter }TransitiveClosureBinaryRelation}}{475}{subsection.33.4.3}
\contentsline {subsection}{\numberline {33.4.4}\leavevmode {\color {Chapter }HasseDiagramBinaryRelation}}{475}{subsection.33.4.4}
\contentsline {subsection}{\numberline {33.4.5}\leavevmode {\color {Chapter }StronglyConnectedComponents}}{475}{subsection.33.4.5}
\contentsline {subsection}{\numberline {33.4.6}\leavevmode {\color {Chapter }PartialOrderByOrderingFunction}}{475}{subsection.33.4.6}
\contentsline {section}{\numberline {33.5}\leavevmode {\color {Chapter }Equivalence Relations}}{475}{section.33.5}
\contentsline {subsection}{\numberline {33.5.1}\leavevmode {\color {Chapter }EquivalenceRelationByPartition}}{476}{subsection.33.5.1}
\contentsline {subsection}{\numberline {33.5.2}\leavevmode {\color {Chapter }EquivalenceRelationByRelation}}{476}{subsection.33.5.2}
\contentsline {subsection}{\numberline {33.5.3}\leavevmode {\color {Chapter }EquivalenceRelationByPairs}}{476}{subsection.33.5.3}
\contentsline {subsection}{\numberline {33.5.4}\leavevmode {\color {Chapter }EquivalenceRelationByProperty}}{476}{subsection.33.5.4}
\contentsline {section}{\numberline {33.6}\leavevmode {\color {Chapter }Attributes of and Operations on Equivalence Relations}}{477}{section.33.6}
\contentsline {subsection}{\numberline {33.6.1}\leavevmode {\color {Chapter }EquivalenceRelationPartition}}{477}{subsection.33.6.1}
\contentsline {subsection}{\numberline {33.6.2}\leavevmode {\color {Chapter }GeneratorsOfEquivalenceRelationPartition}}{477}{subsection.33.6.2}
\contentsline {subsection}{\numberline {33.6.3}\leavevmode {\color {Chapter }JoinEquivalenceRelations}}{477}{subsection.33.6.3}
\contentsline {section}{\numberline {33.7}\leavevmode {\color {Chapter }Equivalence Classes}}{477}{section.33.7}
\contentsline {subsection}{\numberline {33.7.1}\leavevmode {\color {Chapter }IsEquivalenceClass}}{477}{subsection.33.7.1}
\contentsline {subsection}{\numberline {33.7.2}\leavevmode {\color {Chapter }EquivalenceClassRelation}}{477}{subsection.33.7.2}
\contentsline {subsection}{\numberline {33.7.3}\leavevmode {\color {Chapter }EquivalenceClasses (attribute)}}{478}{subsection.33.7.3}
\contentsline {subsection}{\numberline {33.7.4}\leavevmode {\color {Chapter }EquivalenceClassOfElement}}{478}{subsection.33.7.4}
\contentsline {chapter}{\numberline {34}\leavevmode {\color {Chapter }Orderings}}{479}{chapter.34}
\contentsline {section}{\numberline {34.1}\leavevmode {\color {Chapter }IsOrdering (Filter)}}{479}{section.34.1}
\contentsline {subsection}{\numberline {34.1.1}\leavevmode {\color {Chapter }IsOrdering}}{479}{subsection.34.1.1}
\contentsline {subsection}{\numberline {34.1.2}\leavevmode {\color {Chapter }OrderingsFamily}}{479}{subsection.34.1.2}
\contentsline {section}{\numberline {34.2}\leavevmode {\color {Chapter }Building new orderings}}{479}{section.34.2}
\contentsline {subsection}{\numberline {34.2.1}\leavevmode {\color {Chapter }OrderingByLessThanFunctionNC}}{479}{subsection.34.2.1}
\contentsline {subsection}{\numberline {34.2.2}\leavevmode {\color {Chapter }OrderingByLessThanOrEqualFunctionNC}}{479}{subsection.34.2.2}
\contentsline {section}{\numberline {34.3}\leavevmode {\color {Chapter }Properties and basic functionality}}{480}{section.34.3}
\contentsline {subsection}{\numberline {34.3.1}\leavevmode {\color {Chapter }IsWellFoundedOrdering}}{480}{subsection.34.3.1}
\contentsline {subsection}{\numberline {34.3.2}\leavevmode {\color {Chapter }IsTotalOrdering}}{480}{subsection.34.3.2}
\contentsline {subsection}{\numberline {34.3.3}\leavevmode {\color {Chapter }IsIncomparableUnder}}{480}{subsection.34.3.3}
\contentsline {subsection}{\numberline {34.3.4}\leavevmode {\color {Chapter }FamilyForOrdering}}{481}{subsection.34.3.4}
\contentsline {subsection}{\numberline {34.3.5}\leavevmode {\color {Chapter }LessThanFunction}}{481}{subsection.34.3.5}
\contentsline {subsection}{\numberline {34.3.6}\leavevmode {\color {Chapter }LessThanOrEqualFunction}}{481}{subsection.34.3.6}
\contentsline {subsection}{\numberline {34.3.7}\leavevmode {\color {Chapter }IsLessThanUnder}}{481}{subsection.34.3.7}
\contentsline {subsection}{\numberline {34.3.8}\leavevmode {\color {Chapter }IsLessThanOrEqualUnder}}{481}{subsection.34.3.8}
\contentsline {section}{\numberline {34.4}\leavevmode {\color {Chapter }Orderings on families of associative words}}{481}{section.34.4}
\contentsline {subsection}{\numberline {34.4.1}\leavevmode {\color {Chapter }IsOrderingOnFamilyOfAssocWords}}{482}{subsection.34.4.1}
\contentsline {subsection}{\numberline {34.4.2}\leavevmode {\color {Chapter }IsTranslationInvariantOrdering}}{482}{subsection.34.4.2}
\contentsline {subsection}{\numberline {34.4.3}\leavevmode {\color {Chapter }IsReductionOrdering}}{482}{subsection.34.4.3}
\contentsline {subsection}{\numberline {34.4.4}\leavevmode {\color {Chapter }OrderingOnGenerators}}{482}{subsection.34.4.4}
\contentsline {subsection}{\numberline {34.4.5}\leavevmode {\color {Chapter }LexicographicOrdering}}{482}{subsection.34.4.5}
\contentsline {subsection}{\numberline {34.4.6}\leavevmode {\color {Chapter }ShortLexOrdering}}{483}{subsection.34.4.6}
\contentsline {subsection}{\numberline {34.4.7}\leavevmode {\color {Chapter }IsShortLexOrdering}}{483}{subsection.34.4.7}
\contentsline {subsection}{\numberline {34.4.8}\leavevmode {\color {Chapter }WeightLexOrdering}}{483}{subsection.34.4.8}
\contentsline {subsection}{\numberline {34.4.9}\leavevmode {\color {Chapter }IsWeightLexOrdering}}{484}{subsection.34.4.9}
\contentsline {subsection}{\numberline {34.4.10}\leavevmode {\color {Chapter }WeightOfGenerators}}{484}{subsection.34.4.10}
\contentsline {subsection}{\numberline {34.4.11}\leavevmode {\color {Chapter }BasicWreathProductOrdering}}{484}{subsection.34.4.11}
\contentsline {subsection}{\numberline {34.4.12}\leavevmode {\color {Chapter }IsBasicWreathProductOrdering}}{484}{subsection.34.4.12}
\contentsline {subsection}{\numberline {34.4.13}\leavevmode {\color {Chapter }WreathProductOrdering}}{485}{subsection.34.4.13}
\contentsline {subsection}{\numberline {34.4.14}\leavevmode {\color {Chapter }IsWreathProductOrdering}}{485}{subsection.34.4.14}
\contentsline {subsection}{\numberline {34.4.15}\leavevmode {\color {Chapter }LevelsOfGenerators}}{485}{subsection.34.4.15}
\contentsline {chapter}{\numberline {35}\leavevmode {\color {Chapter }Magmas}}{486}{chapter.35}
\contentsline {section}{\numberline {35.1}\leavevmode {\color {Chapter }Magma Categories}}{486}{section.35.1}
\contentsline {subsection}{\numberline {35.1.1}\leavevmode {\color {Chapter }IsMagma}}{486}{subsection.35.1.1}
\contentsline {subsection}{\numberline {35.1.2}\leavevmode {\color {Chapter }IsMagmaWithOne}}{486}{subsection.35.1.2}
\contentsline {subsection}{\numberline {35.1.3}\leavevmode {\color {Chapter }IsMagmaWithInversesIfNonzero}}{486}{subsection.35.1.3}
\contentsline {subsection}{\numberline {35.1.4}\leavevmode {\color {Chapter }IsMagmaWithInverses}}{487}{subsection.35.1.4}
\contentsline {section}{\numberline {35.2}\leavevmode {\color {Chapter }Magma Generation}}{487}{section.35.2}
\contentsline {subsection}{\numberline {35.2.1}\leavevmode {\color {Chapter }Magma}}{487}{subsection.35.2.1}
\contentsline {subsection}{\numberline {35.2.2}\leavevmode {\color {Chapter }MagmaWithOne}}{487}{subsection.35.2.2}
\contentsline {subsection}{\numberline {35.2.3}\leavevmode {\color {Chapter }MagmaWithInverses}}{488}{subsection.35.2.3}
\contentsline {subsection}{\numberline {35.2.4}\leavevmode {\color {Chapter }MagmaByGenerators}}{488}{subsection.35.2.4}
\contentsline {subsection}{\numberline {35.2.5}\leavevmode {\color {Chapter }MagmaWithOneByGenerators}}{488}{subsection.35.2.5}
\contentsline {subsection}{\numberline {35.2.6}\leavevmode {\color {Chapter }MagmaWithInversesByGenerators}}{488}{subsection.35.2.6}
\contentsline {subsection}{\numberline {35.2.7}\leavevmode {\color {Chapter }Submagma}}{488}{subsection.35.2.7}
\contentsline {subsection}{\numberline {35.2.8}\leavevmode {\color {Chapter }SubmagmaWithOne}}{488}{subsection.35.2.8}
\contentsline {subsection}{\numberline {35.2.9}\leavevmode {\color {Chapter }SubmagmaWithInverses}}{488}{subsection.35.2.9}
\contentsline {subsection}{\numberline {35.2.10}\leavevmode {\color {Chapter }AsMagma}}{489}{subsection.35.2.10}
\contentsline {subsection}{\numberline {35.2.11}\leavevmode {\color {Chapter }AsSubmagma}}{489}{subsection.35.2.11}
\contentsline {subsection}{\numberline {35.2.12}\leavevmode {\color {Chapter }IsMagmaWithZeroAdjoined}}{489}{subsection.35.2.12}
\contentsline {subsection}{\numberline {35.2.13}\leavevmode {\color {Chapter }InjectionZeroMagma}}{489}{subsection.35.2.13}
\contentsline {subsection}{\numberline {35.2.14}\leavevmode {\color {Chapter }UnderlyingInjectionZeroMagma}}{490}{subsection.35.2.14}
\contentsline {section}{\numberline {35.3}\leavevmode {\color {Chapter }Magmas Defined by Multiplication Tables}}{490}{section.35.3}
\contentsline {subsection}{\numberline {35.3.1}\leavevmode {\color {Chapter }MagmaByMultiplicationTable}}{490}{subsection.35.3.1}
\contentsline {subsection}{\numberline {35.3.2}\leavevmode {\color {Chapter }MagmaWithOneByMultiplicationTable}}{491}{subsection.35.3.2}
\contentsline {subsection}{\numberline {35.3.3}\leavevmode {\color {Chapter }MagmaWithInversesByMultiplicationTable}}{491}{subsection.35.3.3}
\contentsline {subsection}{\numberline {35.3.4}\leavevmode {\color {Chapter }MagmaElement}}{491}{subsection.35.3.4}
\contentsline {subsection}{\numberline {35.3.5}\leavevmode {\color {Chapter }MultiplicationTable}}{491}{subsection.35.3.5}
\contentsline {section}{\numberline {35.4}\leavevmode {\color {Chapter }Attributes and Properties for Magmas}}{492}{section.35.4}
\contentsline {subsection}{\numberline {35.4.1}\leavevmode {\color {Chapter }GeneratorsOfMagma}}{492}{subsection.35.4.1}
\contentsline {subsection}{\numberline {35.4.2}\leavevmode {\color {Chapter }GeneratorsOfMagmaWithOne}}{492}{subsection.35.4.2}
\contentsline {subsection}{\numberline {35.4.3}\leavevmode {\color {Chapter }GeneratorsOfMagmaWithInverses}}{493}{subsection.35.4.3}
\contentsline {subsection}{\numberline {35.4.4}\leavevmode {\color {Chapter }Centralizer (for a magma and an element)}}{493}{subsection.35.4.4}
\contentsline {subsection}{\numberline {35.4.5}\leavevmode {\color {Chapter }Centre}}{493}{subsection.35.4.5}
\contentsline {subsection}{\numberline {35.4.6}\leavevmode {\color {Chapter }Idempotents}}{493}{subsection.35.4.6}
\contentsline {subsection}{\numberline {35.4.7}\leavevmode {\color {Chapter }IsAssociative}}{494}{subsection.35.4.7}
\contentsline {subsection}{\numberline {35.4.8}\leavevmode {\color {Chapter }IsCentral}}{494}{subsection.35.4.8}
\contentsline {subsection}{\numberline {35.4.9}\leavevmode {\color {Chapter }IsCommutative}}{494}{subsection.35.4.9}
\contentsline {subsection}{\numberline {35.4.10}\leavevmode {\color {Chapter }MultiplicativeNeutralElement}}{494}{subsection.35.4.10}
\contentsline {subsection}{\numberline {35.4.11}\leavevmode {\color {Chapter }MultiplicativeZero}}{494}{subsection.35.4.11}
\contentsline {subsection}{\numberline {35.4.12}\leavevmode {\color {Chapter }SquareRoots}}{495}{subsection.35.4.12}
\contentsline {subsection}{\numberline {35.4.13}\leavevmode {\color {Chapter }TrivialSubmagmaWithOne}}{495}{subsection.35.4.13}
\contentsline {chapter}{\numberline {36}\leavevmode {\color {Chapter }Words}}{496}{chapter.36}
\contentsline {section}{\numberline {36.1}\leavevmode {\color {Chapter }Categories of Words and Nonassociative Words}}{496}{section.36.1}
\contentsline {subsection}{\numberline {36.1.1}\leavevmode {\color {Chapter }IsWord}}{496}{subsection.36.1.1}
\contentsline {subsection}{\numberline {36.1.2}\leavevmode {\color {Chapter }IsWordCollection}}{497}{subsection.36.1.2}
\contentsline {subsection}{\numberline {36.1.3}\leavevmode {\color {Chapter }IsNonassocWord}}{497}{subsection.36.1.3}
\contentsline {subsection}{\numberline {36.1.4}\leavevmode {\color {Chapter }IsNonassocWordCollection}}{498}{subsection.36.1.4}
\contentsline {section}{\numberline {36.2}\leavevmode {\color {Chapter }Comparison of Words}}{498}{section.36.2}
\contentsline {subsection}{\numberline {36.2.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= (for nonassociative words)}}{498}{subsection.36.2.1}
\contentsline {subsection}{\numberline {36.2.2}\leavevmode {\color {Chapter }\texttt {\char 92\relax }{\textless } (for nonassociative words)}}{498}{subsection.36.2.2}
\contentsline {section}{\numberline {36.3}\leavevmode {\color {Chapter }Operations for Words}}{499}{section.36.3}
\contentsline {subsection}{\numberline {36.3.1}\leavevmode {\color {Chapter }MappedWord}}{499}{subsection.36.3.1}
\contentsline {section}{\numberline {36.4}\leavevmode {\color {Chapter }Free Magmas}}{500}{section.36.4}
\contentsline {subsection}{\numberline {36.4.1}\leavevmode {\color {Chapter }FreeMagma}}{500}{subsection.36.4.1}
\contentsline {subsection}{\numberline {36.4.2}\leavevmode {\color {Chapter }FreeMagmaWithOne}}{500}{subsection.36.4.2}
\contentsline {section}{\numberline {36.5}\leavevmode {\color {Chapter }External Representation for Nonassociative Words}}{501}{section.36.5}
\contentsline {chapter}{\numberline {37}\leavevmode {\color {Chapter }Associative Words}}{502}{chapter.37}
\contentsline {section}{\numberline {37.1}\leavevmode {\color {Chapter }Categories of Associative Words}}{502}{section.37.1}
\contentsline {subsection}{\numberline {37.1.1}\leavevmode {\color {Chapter }IsAssocWord}}{503}{subsection.37.1.1}
\contentsline {section}{\numberline {37.2}\leavevmode {\color {Chapter }Free Groups, Monoids and Semigroups}}{503}{section.37.2}
\contentsline {subsection}{\numberline {37.2.1}\leavevmode {\color {Chapter }FreeGroup}}{503}{subsection.37.2.1}
\contentsline {subsection}{\numberline {37.2.2}\leavevmode {\color {Chapter }IsFreeGroup}}{503}{subsection.37.2.2}
\contentsline {subsection}{\numberline {37.2.3}\leavevmode {\color {Chapter }AssignGeneratorVariables}}{504}{subsection.37.2.3}
\contentsline {section}{\numberline {37.3}\leavevmode {\color {Chapter }Comparison of Associative Words}}{504}{section.37.3}
\contentsline {subsection}{\numberline {37.3.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= (for associative words)}}{504}{subsection.37.3.1}
\contentsline {subsection}{\numberline {37.3.2}\leavevmode {\color {Chapter }\texttt {\char 92\relax }{\textless } (for associative words)}}{505}{subsection.37.3.2}
\contentsline {subsection}{\numberline {37.3.3}\leavevmode {\color {Chapter }IsShortLexLessThanOrEqual}}{505}{subsection.37.3.3}
\contentsline {subsection}{\numberline {37.3.4}\leavevmode {\color {Chapter }IsBasicWreathLessThanOrEqual}}{505}{subsection.37.3.4}
\contentsline {section}{\numberline {37.4}\leavevmode {\color {Chapter }Operations for Associative Words}}{505}{section.37.4}
\contentsline {subsection}{\numberline {37.4.1}\leavevmode {\color {Chapter }Length (for a associative word)}}{506}{subsection.37.4.1}
\contentsline {subsection}{\numberline {37.4.2}\leavevmode {\color {Chapter }ExponentSumWord}}{506}{subsection.37.4.2}
\contentsline {subsection}{\numberline {37.4.3}\leavevmode {\color {Chapter }Subword}}{506}{subsection.37.4.3}
\contentsline {subsection}{\numberline {37.4.4}\leavevmode {\color {Chapter }PositionWord}}{506}{subsection.37.4.4}
\contentsline {subsection}{\numberline {37.4.5}\leavevmode {\color {Chapter }SubstitutedWord}}{507}{subsection.37.4.5}
\contentsline {subsection}{\numberline {37.4.6}\leavevmode {\color {Chapter }EliminatedWord}}{507}{subsection.37.4.6}
\contentsline {section}{\numberline {37.5}\leavevmode {\color {Chapter }Operations for Associative Words by their Syllables}}{508}{section.37.5}
\contentsline {subsection}{\numberline {37.5.1}\leavevmode {\color {Chapter }NumberSyllables}}{508}{subsection.37.5.1}
\contentsline {subsection}{\numberline {37.5.2}\leavevmode {\color {Chapter }ExponentSyllable}}{508}{subsection.37.5.2}
\contentsline {subsection}{\numberline {37.5.3}\leavevmode {\color {Chapter }GeneratorSyllable}}{508}{subsection.37.5.3}
\contentsline {subsection}{\numberline {37.5.4}\leavevmode {\color {Chapter }SubSyllables}}{508}{subsection.37.5.4}
\contentsline {section}{\numberline {37.6}\leavevmode {\color {Chapter }Representations for Associative Words}}{509}{section.37.6}
\contentsline {subsection}{\numberline {37.6.1}\leavevmode {\color {Chapter }IsLetterAssocWordRep}}{509}{subsection.37.6.1}
\contentsline {subsection}{\numberline {37.6.2}\leavevmode {\color {Chapter }IsLetterWordsFamily}}{509}{subsection.37.6.2}
\contentsline {subsection}{\numberline {37.6.3}\leavevmode {\color {Chapter }IsBLetterAssocWordRep}}{509}{subsection.37.6.3}
\contentsline {subsection}{\numberline {37.6.4}\leavevmode {\color {Chapter }IsBLetterWordsFamily}}{509}{subsection.37.6.4}
\contentsline {subsection}{\numberline {37.6.5}\leavevmode {\color {Chapter }IsSyllableAssocWordRep}}{510}{subsection.37.6.5}
\contentsline {subsection}{\numberline {37.6.6}\leavevmode {\color {Chapter }IsSyllableWordsFamily}}{510}{subsection.37.6.6}
\contentsline {subsection}{\numberline {37.6.7}\leavevmode {\color {Chapter }Is16BitsFamily}}{510}{subsection.37.6.7}
\contentsline {subsection}{\numberline {37.6.8}\leavevmode {\color {Chapter }LetterRepAssocWord}}{510}{subsection.37.6.8}
\contentsline {subsection}{\numberline {37.6.9}\leavevmode {\color {Chapter }AssocWordByLetterRep}}{510}{subsection.37.6.9}
\contentsline {section}{\numberline {37.7}\leavevmode {\color {Chapter }The External Representation for Associative Words}}{511}{section.37.7}
\contentsline {section}{\numberline {37.8}\leavevmode {\color {Chapter }Straight Line Programs}}{511}{section.37.8}
\contentsline {subsection}{\numberline {37.8.1}\leavevmode {\color {Chapter }IsStraightLineProgram}}{512}{subsection.37.8.1}
\contentsline {subsection}{\numberline {37.8.2}\leavevmode {\color {Chapter }StraightLineProgram (for a list of lines (and the number of generators))}}{512}{subsection.37.8.2}
\contentsline {subsection}{\numberline {37.8.3}\leavevmode {\color {Chapter }LinesOfStraightLineProgram}}{512}{subsection.37.8.3}
\contentsline {subsection}{\numberline {37.8.4}\leavevmode {\color {Chapter }NrInputsOfStraightLineProgram}}{512}{subsection.37.8.4}
\contentsline {subsection}{\numberline {37.8.5}\leavevmode {\color {Chapter }ResultOfStraightLineProgram}}{513}{subsection.37.8.5}
\contentsline {subsection}{\numberline {37.8.6}\leavevmode {\color {Chapter }StringOfResultOfStraightLineProgram}}{514}{subsection.37.8.6}
\contentsline {subsection}{\numberline {37.8.7}\leavevmode {\color {Chapter }CompositionOfStraightLinePrograms}}{515}{subsection.37.8.7}
\contentsline {subsection}{\numberline {37.8.8}\leavevmode {\color {Chapter }IntegratedStraightLineProgram}}{515}{subsection.37.8.8}
\contentsline {subsection}{\numberline {37.8.9}\leavevmode {\color {Chapter }RestrictOutputsOfSLP}}{516}{subsection.37.8.9}
\contentsline {subsection}{\numberline {37.8.10}\leavevmode {\color {Chapter }IntermediateResultOfSLP}}{516}{subsection.37.8.10}
\contentsline {subsection}{\numberline {37.8.11}\leavevmode {\color {Chapter }IntermediateResultOfSLPWithoutOverwrite}}{516}{subsection.37.8.11}
\contentsline {subsection}{\numberline {37.8.12}\leavevmode {\color {Chapter }IntermediateResultsOfSLPWithoutOverwrite}}{516}{subsection.37.8.12}
\contentsline {subsection}{\numberline {37.8.13}\leavevmode {\color {Chapter }ProductOfStraightLinePrograms}}{516}{subsection.37.8.13}
\contentsline {subsection}{\numberline {37.8.14}\leavevmode {\color {Chapter }SlotUsagePattern}}{517}{subsection.37.8.14}
\contentsline {section}{\numberline {37.9}\leavevmode {\color {Chapter }Straight Line Program Elements}}{517}{section.37.9}
\contentsline {subsection}{\numberline {37.9.1}\leavevmode {\color {Chapter }IsStraightLineProgElm}}{517}{subsection.37.9.1}
\contentsline {subsection}{\numberline {37.9.2}\leavevmode {\color {Chapter }StraightLineProgElm}}{517}{subsection.37.9.2}
\contentsline {subsection}{\numberline {37.9.3}\leavevmode {\color {Chapter }StraightLineProgGens}}{518}{subsection.37.9.3}
\contentsline {subsection}{\numberline {37.9.4}\leavevmode {\color {Chapter }EvalStraightLineProgElm}}{518}{subsection.37.9.4}
\contentsline {subsection}{\numberline {37.9.5}\leavevmode {\color {Chapter }StretchImportantSLPElement}}{518}{subsection.37.9.5}
\contentsline {chapter}{\numberline {38}\leavevmode {\color {Chapter }Rewriting Systems}}{519}{chapter.38}
\contentsline {section}{\numberline {38.1}\leavevmode {\color {Chapter }Operations on rewriting systems}}{519}{section.38.1}
\contentsline {subsection}{\numberline {38.1.1}\leavevmode {\color {Chapter }IsRewritingSystem}}{519}{subsection.38.1.1}
\contentsline {subsection}{\numberline {38.1.2}\leavevmode {\color {Chapter }Rules}}{519}{subsection.38.1.2}
\contentsline {subsection}{\numberline {38.1.3}\leavevmode {\color {Chapter }OrderOfRewritingSystem}}{520}{subsection.38.1.3}
\contentsline {subsection}{\numberline {38.1.4}\leavevmode {\color {Chapter }ReducedForm}}{520}{subsection.38.1.4}
\contentsline {subsection}{\numberline {38.1.5}\leavevmode {\color {Chapter }IsConfluent}}{520}{subsection.38.1.5}
\contentsline {subsection}{\numberline {38.1.6}\leavevmode {\color {Chapter }ConfluentRws}}{520}{subsection.38.1.6}
\contentsline {subsection}{\numberline {38.1.7}\leavevmode {\color {Chapter }IsReduced}}{520}{subsection.38.1.7}
\contentsline {subsection}{\numberline {38.1.8}\leavevmode {\color {Chapter }ReduceRules}}{520}{subsection.38.1.8}
\contentsline {subsection}{\numberline {38.1.9}\leavevmode {\color {Chapter }AddRule}}{521}{subsection.38.1.9}
\contentsline {subsection}{\numberline {38.1.10}\leavevmode {\color {Chapter }AddRuleReduced}}{521}{subsection.38.1.10}
\contentsline {subsection}{\numberline {38.1.11}\leavevmode {\color {Chapter }MakeConfluent}}{521}{subsection.38.1.11}
\contentsline {subsection}{\numberline {38.1.12}\leavevmode {\color {Chapter }GeneratorsOfRws}}{521}{subsection.38.1.12}
\contentsline {section}{\numberline {38.2}\leavevmode {\color {Chapter }Operations on elements of the algebra}}{521}{section.38.2}
\contentsline {subsection}{\numberline {38.2.1}\leavevmode {\color {Chapter }ReducedProduct}}{521}{subsection.38.2.1}
\contentsline {section}{\numberline {38.3}\leavevmode {\color {Chapter }Properties of rewriting systems}}{522}{section.38.3}
\contentsline {subsection}{\numberline {38.3.1}\leavevmode {\color {Chapter }IsBuiltFromAdditiveMagmaWithInverses}}{522}{subsection.38.3.1}
\contentsline {section}{\numberline {38.4}\leavevmode {\color {Chapter }Rewriting in Groups and Monoids}}{522}{section.38.4}
\contentsline {section}{\numberline {38.5}\leavevmode {\color {Chapter }Developing rewriting systems}}{523}{section.38.5}
\contentsline {chapter}{\numberline {39}\leavevmode {\color {Chapter }Groups}}{525}{chapter.39}
\contentsline {section}{\numberline {39.1}\leavevmode {\color {Chapter }Group Elements}}{525}{section.39.1}
\contentsline {section}{\numberline {39.2}\leavevmode {\color {Chapter }Creating Groups}}{526}{section.39.2}
\contentsline {subsection}{\numberline {39.2.1}\leavevmode {\color {Chapter }Group (for several generators)}}{526}{subsection.39.2.1}
\contentsline {subsection}{\numberline {39.2.2}\leavevmode {\color {Chapter }GroupByGenerators}}{526}{subsection.39.2.2}
\contentsline {subsection}{\numberline {39.2.3}\leavevmode {\color {Chapter }GroupWithGenerators}}{526}{subsection.39.2.3}
\contentsline {subsection}{\numberline {39.2.4}\leavevmode {\color {Chapter }GeneratorsOfGroup}}{526}{subsection.39.2.4}
\contentsline {subsection}{\numberline {39.2.5}\leavevmode {\color {Chapter }AsGroup}}{527}{subsection.39.2.5}
\contentsline {subsection}{\numberline {39.2.6}\leavevmode {\color {Chapter }ConjugateGroup}}{527}{subsection.39.2.6}
\contentsline {subsection}{\numberline {39.2.7}\leavevmode {\color {Chapter }IsGroup}}{527}{subsection.39.2.7}
\contentsline {subsection}{\numberline {39.2.8}\leavevmode {\color {Chapter }InfoGroup}}{528}{subsection.39.2.8}
\contentsline {section}{\numberline {39.3}\leavevmode {\color {Chapter }Subgroups}}{528}{section.39.3}
\contentsline {subsection}{\numberline {39.3.1}\leavevmode {\color {Chapter }Subgroup}}{528}{subsection.39.3.1}
\contentsline {subsection}{\numberline {39.3.2}\leavevmode {\color {Chapter }Index (\textsf {GAP} operation)}}{528}{subsection.39.3.2}
\contentsline {subsection}{\numberline {39.3.3}\leavevmode {\color {Chapter }IndexInWholeGroup}}{528}{subsection.39.3.3}
\contentsline {subsection}{\numberline {39.3.4}\leavevmode {\color {Chapter }AsSubgroup}}{529}{subsection.39.3.4}
\contentsline {subsection}{\numberline {39.3.5}\leavevmode {\color {Chapter }IsSubgroup}}{529}{subsection.39.3.5}
\contentsline {subsection}{\numberline {39.3.6}\leavevmode {\color {Chapter }IsNormal}}{529}{subsection.39.3.6}
\contentsline {subsection}{\numberline {39.3.7}\leavevmode {\color {Chapter }IsCharacteristicSubgroup}}{530}{subsection.39.3.7}
\contentsline {subsection}{\numberline {39.3.8}\leavevmode {\color {Chapter }ConjugateSubgroup}}{530}{subsection.39.3.8}
\contentsline {subsection}{\numberline {39.3.9}\leavevmode {\color {Chapter }ConjugateSubgroups}}{530}{subsection.39.3.9}
\contentsline {subsection}{\numberline {39.3.10}\leavevmode {\color {Chapter }IsSubnormal}}{530}{subsection.39.3.10}
\contentsline {subsection}{\numberline {39.3.11}\leavevmode {\color {Chapter }SubgroupByProperty}}{530}{subsection.39.3.11}
\contentsline {subsection}{\numberline {39.3.12}\leavevmode {\color {Chapter }SubgroupShell}}{531}{subsection.39.3.12}
\contentsline {section}{\numberline {39.4}\leavevmode {\color {Chapter }Closures of (Sub)groups}}{531}{section.39.4}
\contentsline {subsection}{\numberline {39.4.1}\leavevmode {\color {Chapter }ClosureGroup}}{531}{subsection.39.4.1}
\contentsline {subsection}{\numberline {39.4.2}\leavevmode {\color {Chapter }ClosureGroupAddElm}}{531}{subsection.39.4.2}
\contentsline {subsection}{\numberline {39.4.3}\leavevmode {\color {Chapter }ClosureGroupDefault}}{531}{subsection.39.4.3}
\contentsline {subsection}{\numberline {39.4.4}\leavevmode {\color {Chapter }ClosureSubgroup}}{532}{subsection.39.4.4}
\contentsline {section}{\numberline {39.5}\leavevmode {\color {Chapter }Expressing Group Elements as Words in Generators}}{532}{section.39.5}
\contentsline {subsection}{\numberline {39.5.1}\leavevmode {\color {Chapter }EpimorphismFromFreeGroup}}{532}{subsection.39.5.1}
\contentsline {subsection}{\numberline {39.5.2}\leavevmode {\color {Chapter }Factorization}}{533}{subsection.39.5.2}
\contentsline {subsection}{\numberline {39.5.3}\leavevmode {\color {Chapter }GrowthFunctionOfGroup}}{533}{subsection.39.5.3}
\contentsline {section}{\numberline {39.6}\leavevmode {\color {Chapter }Structure Descriptions}}{534}{section.39.6}
\contentsline {subsection}{\numberline {39.6.1}\leavevmode {\color {Chapter }StructureDescription}}{534}{subsection.39.6.1}
\contentsline {section}{\numberline {39.7}\leavevmode {\color {Chapter }Cosets}}{536}{section.39.7}
\contentsline {subsection}{\numberline {39.7.1}\leavevmode {\color {Chapter }RightCoset}}{536}{subsection.39.7.1}
\contentsline {subsection}{\numberline {39.7.2}\leavevmode {\color {Chapter }RightCosets}}{537}{subsection.39.7.2}
\contentsline {subsection}{\numberline {39.7.3}\leavevmode {\color {Chapter }CanonicalRightCosetElement}}{537}{subsection.39.7.3}
\contentsline {subsection}{\numberline {39.7.4}\leavevmode {\color {Chapter }IsRightCoset}}{538}{subsection.39.7.4}
\contentsline {subsection}{\numberline {39.7.5}\leavevmode {\color {Chapter }IsBiCoset}}{538}{subsection.39.7.5}
\contentsline {subsection}{\numberline {39.7.6}\leavevmode {\color {Chapter }CosetDecomposition}}{538}{subsection.39.7.6}
\contentsline {section}{\numberline {39.8}\leavevmode {\color {Chapter }Transversals}}{538}{section.39.8}
\contentsline {subsection}{\numberline {39.8.1}\leavevmode {\color {Chapter }RightTransversal}}{538}{subsection.39.8.1}
\contentsline {section}{\numberline {39.9}\leavevmode {\color {Chapter }Double Cosets}}{539}{section.39.9}
\contentsline {subsection}{\numberline {39.9.1}\leavevmode {\color {Chapter }DoubleCoset}}{539}{subsection.39.9.1}
\contentsline {subsection}{\numberline {39.9.2}\leavevmode {\color {Chapter }RepresentativesContainedRightCosets}}{539}{subsection.39.9.2}
\contentsline {subsection}{\numberline {39.9.3}\leavevmode {\color {Chapter }DoubleCosets}}{540}{subsection.39.9.3}
\contentsline {subsection}{\numberline {39.9.4}\leavevmode {\color {Chapter }IsDoubleCoset (operation)}}{540}{subsection.39.9.4}
\contentsline {subsection}{\numberline {39.9.5}\leavevmode {\color {Chapter }DoubleCosetRepsAndSizes}}{540}{subsection.39.9.5}
\contentsline {subsection}{\numberline {39.9.6}\leavevmode {\color {Chapter }InfoCoset}}{540}{subsection.39.9.6}
\contentsline {section}{\numberline {39.10}\leavevmode {\color {Chapter }Conjugacy Classes}}{541}{section.39.10}
\contentsline {subsection}{\numberline {39.10.1}\leavevmode {\color {Chapter }ConjugacyClass}}{541}{subsection.39.10.1}
\contentsline {subsection}{\numberline {39.10.2}\leavevmode {\color {Chapter }ConjugacyClasses (attribute)}}{541}{subsection.39.10.2}
\contentsline {subsection}{\numberline {39.10.3}\leavevmode {\color {Chapter }ConjugacyClassesByRandomSearch}}{542}{subsection.39.10.3}
\contentsline {subsection}{\numberline {39.10.4}\leavevmode {\color {Chapter }ConjugacyClassesByOrbits}}{542}{subsection.39.10.4}
\contentsline {subsection}{\numberline {39.10.5}\leavevmode {\color {Chapter }NrConjugacyClasses}}{542}{subsection.39.10.5}
\contentsline {subsection}{\numberline {39.10.6}\leavevmode {\color {Chapter }RationalClass}}{543}{subsection.39.10.6}
\contentsline {subsection}{\numberline {39.10.7}\leavevmode {\color {Chapter }RationalClasses}}{543}{subsection.39.10.7}
\contentsline {subsection}{\numberline {39.10.8}\leavevmode {\color {Chapter }GaloisGroup (of rational class of a group)}}{543}{subsection.39.10.8}
\contentsline {subsection}{\numberline {39.10.9}\leavevmode {\color {Chapter }IsConjugate}}{543}{subsection.39.10.9}
\contentsline {subsection}{\numberline {39.10.10}\leavevmode {\color {Chapter }NthRootsInGroup}}{544}{subsection.39.10.10}
\contentsline {section}{\numberline {39.11}\leavevmode {\color {Chapter }Normal Structure}}{544}{section.39.11}
\contentsline {subsection}{\numberline {39.11.1}\leavevmode {\color {Chapter }Normalizer}}{544}{subsection.39.11.1}
\contentsline {subsection}{\numberline {39.11.2}\leavevmode {\color {Chapter }Core}}{544}{subsection.39.11.2}
\contentsline {subsection}{\numberline {39.11.3}\leavevmode {\color {Chapter }PCore}}{544}{subsection.39.11.3}
\contentsline {subsection}{\numberline {39.11.4}\leavevmode {\color {Chapter }NormalClosure}}{545}{subsection.39.11.4}
\contentsline {subsection}{\numberline {39.11.5}\leavevmode {\color {Chapter }NormalIntersection}}{545}{subsection.39.11.5}
\contentsline {subsection}{\numberline {39.11.6}\leavevmode {\color {Chapter }ComplementClassesRepresentatives}}{545}{subsection.39.11.6}
\contentsline {subsection}{\numberline {39.11.7}\leavevmode {\color {Chapter }InfoComplement}}{546}{subsection.39.11.7}
\contentsline {section}{\numberline {39.12}\leavevmode {\color {Chapter }Specific and Parametrized Subgroups}}{546}{section.39.12}
\contentsline {subsection}{\numberline {39.12.1}\leavevmode {\color {Chapter }TrivialSubgroup}}{546}{subsection.39.12.1}
\contentsline {subsection}{\numberline {39.12.2}\leavevmode {\color {Chapter }CommutatorSubgroup}}{546}{subsection.39.12.2}
\contentsline {subsection}{\numberline {39.12.3}\leavevmode {\color {Chapter }DerivedSubgroup}}{546}{subsection.39.12.3}
\contentsline {subsection}{\numberline {39.12.4}\leavevmode {\color {Chapter }CommutatorLength}}{546}{subsection.39.12.4}
\contentsline {subsection}{\numberline {39.12.5}\leavevmode {\color {Chapter }FittingSubgroup}}{547}{subsection.39.12.5}
\contentsline {subsection}{\numberline {39.12.6}\leavevmode {\color {Chapter }FrattiniSubgroup}}{547}{subsection.39.12.6}
\contentsline {subsection}{\numberline {39.12.7}\leavevmode {\color {Chapter }PrefrattiniSubgroup}}{547}{subsection.39.12.7}
\contentsline {subsection}{\numberline {39.12.8}\leavevmode {\color {Chapter }PerfectResiduum}}{548}{subsection.39.12.8}
\contentsline {subsection}{\numberline {39.12.9}\leavevmode {\color {Chapter }RadicalGroup}}{548}{subsection.39.12.9}
\contentsline {subsection}{\numberline {39.12.10}\leavevmode {\color {Chapter }Socle}}{548}{subsection.39.12.10}
\contentsline {subsection}{\numberline {39.12.11}\leavevmode {\color {Chapter }SupersolvableResiduum}}{548}{subsection.39.12.11}
\contentsline {subsection}{\numberline {39.12.12}\leavevmode {\color {Chapter }PRump}}{548}{subsection.39.12.12}
\contentsline {section}{\numberline {39.13}\leavevmode {\color {Chapter }Sylow Subgroups and Hall Subgroups}}{549}{section.39.13}
\contentsline {subsection}{\numberline {39.13.1}\leavevmode {\color {Chapter }SylowSubgroup}}{549}{subsection.39.13.1}
\contentsline {subsection}{\numberline {39.13.2}\leavevmode {\color {Chapter }SylowComplement}}{549}{subsection.39.13.2}
\contentsline {subsection}{\numberline {39.13.3}\leavevmode {\color {Chapter }HallSubgroup}}{550}{subsection.39.13.3}
\contentsline {subsection}{\numberline {39.13.4}\leavevmode {\color {Chapter }SylowSystem}}{550}{subsection.39.13.4}
\contentsline {subsection}{\numberline {39.13.5}\leavevmode {\color {Chapter }ComplementSystem}}{550}{subsection.39.13.5}
\contentsline {subsection}{\numberline {39.13.6}\leavevmode {\color {Chapter }HallSystem}}{551}{subsection.39.13.6}
\contentsline {section}{\numberline {39.14}\leavevmode {\color {Chapter }Subgroups characterized by prime powers}}{551}{section.39.14}
\contentsline {subsection}{\numberline {39.14.1}\leavevmode {\color {Chapter }Omega}}{551}{subsection.39.14.1}
\contentsline {subsection}{\numberline {39.14.2}\leavevmode {\color {Chapter }Agemo}}{551}{subsection.39.14.2}
\contentsline {section}{\numberline {39.15}\leavevmode {\color {Chapter }Group Properties}}{551}{section.39.15}
\contentsline {subsection}{\numberline {39.15.1}\leavevmode {\color {Chapter }IsCyclic}}{552}{subsection.39.15.1}
\contentsline {subsection}{\numberline {39.15.2}\leavevmode {\color {Chapter }IsElementaryAbelian}}{552}{subsection.39.15.2}
\contentsline {subsection}{\numberline {39.15.3}\leavevmode {\color {Chapter }IsNilpotentGroup}}{552}{subsection.39.15.3}
\contentsline {subsection}{\numberline {39.15.4}\leavevmode {\color {Chapter }NilpotencyClassOfGroup}}{552}{subsection.39.15.4}
\contentsline {subsection}{\numberline {39.15.5}\leavevmode {\color {Chapter }IsPerfectGroup}}{552}{subsection.39.15.5}
\contentsline {subsection}{\numberline {39.15.6}\leavevmode {\color {Chapter }IsSolvableGroup}}{552}{subsection.39.15.6}
\contentsline {subsection}{\numberline {39.15.7}\leavevmode {\color {Chapter }IsPolycyclicGroup}}{553}{subsection.39.15.7}
\contentsline {subsection}{\numberline {39.15.8}\leavevmode {\color {Chapter }IsSupersolvableGroup}}{553}{subsection.39.15.8}
\contentsline {subsection}{\numberline {39.15.9}\leavevmode {\color {Chapter }IsMonomialGroup}}{553}{subsection.39.15.9}
\contentsline {subsection}{\numberline {39.15.10}\leavevmode {\color {Chapter }IsSimpleGroup}}{553}{subsection.39.15.10}
\contentsline {subsection}{\numberline {39.15.11}\leavevmode {\color {Chapter }IsAlmostSimpleGroup}}{553}{subsection.39.15.11}
\contentsline {subsection}{\numberline {39.15.12}\leavevmode {\color {Chapter }IsomorphismTypeInfoFiniteSimpleGroup}}{554}{subsection.39.15.12}
\contentsline {subsection}{\numberline {39.15.13}\leavevmode {\color {Chapter }SimpleGroup}}{555}{subsection.39.15.13}
\contentsline {subsection}{\numberline {39.15.14}\leavevmode {\color {Chapter }SimpleGroupsIterator}}{556}{subsection.39.15.14}
\contentsline {subsection}{\numberline {39.15.15}\leavevmode {\color {Chapter }SmallSimpleGroup}}{556}{subsection.39.15.15}
\contentsline {subsection}{\numberline {39.15.16}\leavevmode {\color {Chapter }AllSmallNonabelianSimpleGroups}}{556}{subsection.39.15.16}
\contentsline {subsection}{\numberline {39.15.17}\leavevmode {\color {Chapter }IsFinitelyGeneratedGroup}}{557}{subsection.39.15.17}
\contentsline {subsection}{\numberline {39.15.18}\leavevmode {\color {Chapter }IsSubsetLocallyFiniteGroup}}{557}{subsection.39.15.18}
\contentsline {subsection}{\numberline {39.15.19}\leavevmode {\color {Chapter }IsPGroup}}{557}{subsection.39.15.19}
\contentsline {subsection}{\numberline {39.15.20}\leavevmode {\color {Chapter }IsPowerfulPGroup}}{557}{subsection.39.15.20}
\contentsline {subsection}{\numberline {39.15.21}\leavevmode {\color {Chapter }PrimePGroup}}{558}{subsection.39.15.21}
\contentsline {subsection}{\numberline {39.15.22}\leavevmode {\color {Chapter }PClassPGroup}}{558}{subsection.39.15.22}
\contentsline {subsection}{\numberline {39.15.23}\leavevmode {\color {Chapter }RankPGroup}}{558}{subsection.39.15.23}
\contentsline {subsection}{\numberline {39.15.24}\leavevmode {\color {Chapter }IsPSolvable}}{558}{subsection.39.15.24}
\contentsline {subsection}{\numberline {39.15.25}\leavevmode {\color {Chapter }IsPNilpotent}}{558}{subsection.39.15.25}
\contentsline {section}{\numberline {39.16}\leavevmode {\color {Chapter }Numerical Group Attributes}}{559}{section.39.16}
\contentsline {subsection}{\numberline {39.16.1}\leavevmode {\color {Chapter }AbelianInvariants}}{559}{subsection.39.16.1}
\contentsline {subsection}{\numberline {39.16.2}\leavevmode {\color {Chapter }Exponent}}{559}{subsection.39.16.2}
\contentsline {subsection}{\numberline {39.16.3}\leavevmode {\color {Chapter }EulerianFunction}}{559}{subsection.39.16.3}
\contentsline {section}{\numberline {39.17}\leavevmode {\color {Chapter }Subgroup Series}}{560}{section.39.17}
\contentsline {subsection}{\numberline {39.17.1}\leavevmode {\color {Chapter }ChiefSeries}}{560}{subsection.39.17.1}
\contentsline {subsection}{\numberline {39.17.2}\leavevmode {\color {Chapter }ChiefSeriesThrough}}{560}{subsection.39.17.2}
\contentsline {subsection}{\numberline {39.17.3}\leavevmode {\color {Chapter }ChiefSeriesUnderAction}}{560}{subsection.39.17.3}
\contentsline {subsection}{\numberline {39.17.4}\leavevmode {\color {Chapter }SubnormalSeries}}{561}{subsection.39.17.4}
\contentsline {subsection}{\numberline {39.17.5}\leavevmode {\color {Chapter }CompositionSeries}}{561}{subsection.39.17.5}
\contentsline {subsection}{\numberline {39.17.6}\leavevmode {\color {Chapter }DisplayCompositionSeries}}{561}{subsection.39.17.6}
\contentsline {subsection}{\numberline {39.17.7}\leavevmode {\color {Chapter }DerivedSeriesOfGroup}}{561}{subsection.39.17.7}
\contentsline {subsection}{\numberline {39.17.8}\leavevmode {\color {Chapter }DerivedLength}}{561}{subsection.39.17.8}
\contentsline {subsection}{\numberline {39.17.9}\leavevmode {\color {Chapter }ElementaryAbelianSeries}}{562}{subsection.39.17.9}
\contentsline {subsection}{\numberline {39.17.10}\leavevmode {\color {Chapter }InvariantElementaryAbelianSeries}}{562}{subsection.39.17.10}
\contentsline {subsection}{\numberline {39.17.11}\leavevmode {\color {Chapter }LowerCentralSeriesOfGroup}}{562}{subsection.39.17.11}
\contentsline {subsection}{\numberline {39.17.12}\leavevmode {\color {Chapter }UpperCentralSeriesOfGroup}}{563}{subsection.39.17.12}
\contentsline {subsection}{\numberline {39.17.13}\leavevmode {\color {Chapter }PCentralSeries}}{563}{subsection.39.17.13}
\contentsline {subsection}{\numberline {39.17.14}\leavevmode {\color {Chapter }JenningsSeries}}{563}{subsection.39.17.14}
\contentsline {subsection}{\numberline {39.17.15}\leavevmode {\color {Chapter }DimensionsLoewyFactors}}{563}{subsection.39.17.15}
\contentsline {subsection}{\numberline {39.17.16}\leavevmode {\color {Chapter }AscendingChain}}{564}{subsection.39.17.16}
\contentsline {subsection}{\numberline {39.17.17}\leavevmode {\color {Chapter }IntermediateGroup}}{564}{subsection.39.17.17}
\contentsline {subsection}{\numberline {39.17.18}\leavevmode {\color {Chapter }IntermediateSubgroups}}{564}{subsection.39.17.18}
\contentsline {section}{\numberline {39.18}\leavevmode {\color {Chapter }Factor Groups}}{564}{section.39.18}
\contentsline {subsection}{\numberline {39.18.1}\leavevmode {\color {Chapter }NaturalHomomorphismByNormalSubgroup}}{564}{subsection.39.18.1}
\contentsline {subsection}{\numberline {39.18.2}\leavevmode {\color {Chapter }FactorGroup}}{564}{subsection.39.18.2}
\contentsline {subsection}{\numberline {39.18.3}\leavevmode {\color {Chapter }CommutatorFactorGroup}}{565}{subsection.39.18.3}
\contentsline {subsection}{\numberline {39.18.4}\leavevmode {\color {Chapter }MaximalAbelianQuotient}}{565}{subsection.39.18.4}
\contentsline {subsection}{\numberline {39.18.5}\leavevmode {\color {Chapter }HasAbelianFactorGroup}}{565}{subsection.39.18.5}
\contentsline {subsection}{\numberline {39.18.6}\leavevmode {\color {Chapter }HasElementaryAbelianFactorGroup}}{565}{subsection.39.18.6}
\contentsline {subsection}{\numberline {39.18.7}\leavevmode {\color {Chapter }CentralizerModulo}}{566}{subsection.39.18.7}
\contentsline {section}{\numberline {39.19}\leavevmode {\color {Chapter }Sets of Subgroups}}{566}{section.39.19}
\contentsline {subsection}{\numberline {39.19.1}\leavevmode {\color {Chapter }ConjugacyClassSubgroups}}{566}{subsection.39.19.1}
\contentsline {subsection}{\numberline {39.19.2}\leavevmode {\color {Chapter }IsConjugacyClassSubgroupsRep}}{566}{subsection.39.19.2}
\contentsline {subsection}{\numberline {39.19.3}\leavevmode {\color {Chapter }ConjugacyClassesSubgroups}}{566}{subsection.39.19.3}
\contentsline {subsection}{\numberline {39.19.4}\leavevmode {\color {Chapter }ConjugacyClassesMaximalSubgroups}}{567}{subsection.39.19.4}
\contentsline {subsection}{\numberline {39.19.5}\leavevmode {\color {Chapter }MaximalSubgroupClassReps}}{567}{subsection.39.19.5}
\contentsline {subsection}{\numberline {39.19.6}\leavevmode {\color {Chapter }LowIndexSubgroups}}{567}{subsection.39.19.6}
\contentsline {subsection}{\numberline {39.19.7}\leavevmode {\color {Chapter }AllSubgroups}}{568}{subsection.39.19.7}
\contentsline {subsection}{\numberline {39.19.8}\leavevmode {\color {Chapter }MaximalSubgroups}}{568}{subsection.39.19.8}
\contentsline {subsection}{\numberline {39.19.9}\leavevmode {\color {Chapter }NormalSubgroups}}{568}{subsection.39.19.9}
\contentsline {subsection}{\numberline {39.19.10}\leavevmode {\color {Chapter }MaximalNormalSubgroups}}{568}{subsection.39.19.10}
\contentsline {subsection}{\numberline {39.19.11}\leavevmode {\color {Chapter }MinimalNormalSubgroups}}{569}{subsection.39.19.11}
\contentsline {subsection}{\numberline {39.19.12}\leavevmode {\color {Chapter }CharacteristicSubgroups}}{569}{subsection.39.19.12}
\contentsline {section}{\numberline {39.20}\leavevmode {\color {Chapter }Subgroup Lattice}}{569}{section.39.20}
\contentsline {subsection}{\numberline {39.20.1}\leavevmode {\color {Chapter }LatticeSubgroups}}{569}{subsection.39.20.1}
\contentsline {subsection}{\numberline {39.20.2}\leavevmode {\color {Chapter }ClassElementLattice}}{570}{subsection.39.20.2}
\contentsline {subsection}{\numberline {39.20.3}\leavevmode {\color {Chapter }DotFileLatticeSubgroups}}{570}{subsection.39.20.3}
\contentsline {subsection}{\numberline {39.20.4}\leavevmode {\color {Chapter }MaximalSubgroupsLattice}}{570}{subsection.39.20.4}
\contentsline {subsection}{\numberline {39.20.5}\leavevmode {\color {Chapter }MinimalSupergroupsLattice}}{571}{subsection.39.20.5}
\contentsline {subsection}{\numberline {39.20.6}\leavevmode {\color {Chapter }LowLayerSubgroups}}{571}{subsection.39.20.6}
\contentsline {subsection}{\numberline {39.20.7}\leavevmode {\color {Chapter }ContainedConjugates}}{572}{subsection.39.20.7}
\contentsline {subsection}{\numberline {39.20.8}\leavevmode {\color {Chapter }ContainingConjugates}}{572}{subsection.39.20.8}
\contentsline {subsection}{\numberline {39.20.9}\leavevmode {\color {Chapter }MinimalFaithfulPermutationDegree}}{572}{subsection.39.20.9}
\contentsline {subsection}{\numberline {39.20.10}\leavevmode {\color {Chapter }RepresentativesPerfectSubgroups}}{572}{subsection.39.20.10}
\contentsline {subsection}{\numberline {39.20.11}\leavevmode {\color {Chapter }ConjugacyClassesPerfectSubgroups}}{573}{subsection.39.20.11}
\contentsline {subsection}{\numberline {39.20.12}\leavevmode {\color {Chapter }Zuppos}}{573}{subsection.39.20.12}
\contentsline {subsection}{\numberline {39.20.13}\leavevmode {\color {Chapter }InfoLattice}}{573}{subsection.39.20.13}
\contentsline {section}{\numberline {39.21}\leavevmode {\color {Chapter }Specific Methods for Subgroup Lattice Computations}}{573}{section.39.21}
\contentsline {subsection}{\numberline {39.21.1}\leavevmode {\color {Chapter }LatticeByCyclicExtension}}{573}{subsection.39.21.1}
\contentsline {subsection}{\numberline {39.21.2}\leavevmode {\color {Chapter }InvariantSubgroupsElementaryAbelianGroup}}{574}{subsection.39.21.2}
\contentsline {subsection}{\numberline {39.21.3}\leavevmode {\color {Chapter }SubgroupsSolvableGroup}}{574}{subsection.39.21.3}
\contentsline {subsection}{\numberline {39.21.4}\leavevmode {\color {Chapter }SizeConsiderFunction}}{576}{subsection.39.21.4}
\contentsline {subsection}{\numberline {39.21.5}\leavevmode {\color {Chapter }ExactSizeConsiderFunction}}{576}{subsection.39.21.5}
\contentsline {subsection}{\numberline {39.21.6}\leavevmode {\color {Chapter }InfoPcSubgroup}}{576}{subsection.39.21.6}
\contentsline {section}{\numberline {39.22}\leavevmode {\color {Chapter }Special Generating Sets}}{577}{section.39.22}
\contentsline {subsection}{\numberline {39.22.1}\leavevmode {\color {Chapter }GeneratorsSmallest}}{577}{subsection.39.22.1}
\contentsline {subsection}{\numberline {39.22.2}\leavevmode {\color {Chapter }LargestElementGroup}}{577}{subsection.39.22.2}
\contentsline {subsection}{\numberline {39.22.3}\leavevmode {\color {Chapter }MinimalGeneratingSet}}{577}{subsection.39.22.3}
\contentsline {subsection}{\numberline {39.22.4}\leavevmode {\color {Chapter }SmallGeneratingSet}}{577}{subsection.39.22.4}
\contentsline {subsection}{\numberline {39.22.5}\leavevmode {\color {Chapter }IndependentGeneratorsOfAbelianGroup}}{578}{subsection.39.22.5}
\contentsline {subsection}{\numberline {39.22.6}\leavevmode {\color {Chapter }IndependentGeneratorExponents}}{578}{subsection.39.22.6}
\contentsline {section}{\numberline {39.23}\leavevmode {\color {Chapter }1-Cohomology}}{578}{section.39.23}
\contentsline {subsection}{\numberline {39.23.1}\leavevmode {\color {Chapter }OneCocycles}}{579}{subsection.39.23.1}
\contentsline {subsection}{\numberline {39.23.2}\leavevmode {\color {Chapter }OneCoboundaries}}{579}{subsection.39.23.2}
\contentsline {subsection}{\numberline {39.23.3}\leavevmode {\color {Chapter }OCOneCocycles}}{580}{subsection.39.23.3}
\contentsline {subsection}{\numberline {39.23.4}\leavevmode {\color {Chapter }ComplementClassesRepresentativesEA}}{581}{subsection.39.23.4}
\contentsline {subsection}{\numberline {39.23.5}\leavevmode {\color {Chapter }InfoCoh}}{581}{subsection.39.23.5}
\contentsline {section}{\numberline {39.24}\leavevmode {\color {Chapter }Schur Covers and Multipliers}}{581}{section.39.24}
\contentsline {subsection}{\numberline {39.24.1}\leavevmode {\color {Chapter }EpimorphismSchurCover}}{581}{subsection.39.24.1}
\contentsline {subsection}{\numberline {39.24.2}\leavevmode {\color {Chapter }SchurCover}}{581}{subsection.39.24.2}
\contentsline {subsection}{\numberline {39.24.3}\leavevmode {\color {Chapter }AbelianInvariantsMultiplier}}{582}{subsection.39.24.3}
\contentsline {subsection}{\numberline {39.24.4}\leavevmode {\color {Chapter }Epicentre}}{582}{subsection.39.24.4}
\contentsline {subsection}{\numberline {39.24.5}\leavevmode {\color {Chapter }NonabelianExteriorSquare}}{583}{subsection.39.24.5}
\contentsline {subsection}{\numberline {39.24.6}\leavevmode {\color {Chapter }EpimorphismNonabelianExteriorSquare}}{583}{subsection.39.24.6}
\contentsline {subsection}{\numberline {39.24.7}\leavevmode {\color {Chapter }IsCentralFactor}}{583}{subsection.39.24.7}
\contentsline {subsection}{\numberline {39.24.8}\leavevmode {\color {Chapter }Covering groups of symmetric groups}}{583}{subsection.39.24.8}
\contentsline {subsection}{\numberline {39.24.9}\leavevmode {\color {Chapter }BasicSpinRepresentationOfSymmetricGroup}}{584}{subsection.39.24.9}
\contentsline {subsection}{\numberline {39.24.10}\leavevmode {\color {Chapter }SchurCoverOfSymmetricGroup}}{584}{subsection.39.24.10}
\contentsline {subsection}{\numberline {39.24.11}\leavevmode {\color {Chapter }DoubleCoverOfAlternatingGroup}}{584}{subsection.39.24.11}
\contentsline {section}{\numberline {39.25}\leavevmode {\color {Chapter }2-Cohomology}}{584}{section.39.25}
\contentsline {subsection}{\numberline {39.25.1}\leavevmode {\color {Chapter }TwoCohomologyGeneric}}{584}{subsection.39.25.1}
\contentsline {subsection}{\numberline {39.25.2}\leavevmode {\color {Chapter }FpGroupCocycle}}{585}{subsection.39.25.2}
\contentsline {section}{\numberline {39.26}\leavevmode {\color {Chapter }Tests for the Availability of Methods}}{586}{section.39.26}
\contentsline {subsection}{\numberline {39.26.1}\leavevmode {\color {Chapter }CanEasilyTestMembership}}{586}{subsection.39.26.1}
\contentsline {subsection}{\numberline {39.26.2}\leavevmode {\color {Chapter }CanEasilyComputeWithIndependentGensAbelianGroup}}{587}{subsection.39.26.2}
\contentsline {subsection}{\numberline {39.26.3}\leavevmode {\color {Chapter }CanComputeSize}}{587}{subsection.39.26.3}
\contentsline {subsection}{\numberline {39.26.4}\leavevmode {\color {Chapter }CanComputeSizeAnySubgroup}}{587}{subsection.39.26.4}
\contentsline {subsection}{\numberline {39.26.5}\leavevmode {\color {Chapter }CanComputeIndex}}{587}{subsection.39.26.5}
\contentsline {subsection}{\numberline {39.26.6}\leavevmode {\color {Chapter }CanComputeIsSubset}}{587}{subsection.39.26.6}
\contentsline {subsection}{\numberline {39.26.7}\leavevmode {\color {Chapter }KnowsHowToDecompose}}{587}{subsection.39.26.7}
\contentsline {section}{\numberline {39.27}\leavevmode {\color {Chapter }Specific functions for Normalizer calculation}}{588}{section.39.27}
\contentsline {subsection}{\numberline {39.27.1}\leavevmode {\color {Chapter }NormalizerViaRadical}}{588}{subsection.39.27.1}
\contentsline {chapter}{\numberline {40}\leavevmode {\color {Chapter }Group Homomorphisms}}{589}{chapter.40}
\contentsline {section}{\numberline {40.1}\leavevmode {\color {Chapter }Creating Group Homomorphisms}}{589}{section.40.1}
\contentsline {subsection}{\numberline {40.1.1}\leavevmode {\color {Chapter }GroupHomomorphismByImages}}{589}{subsection.40.1.1}
\contentsline {subsection}{\numberline {40.1.2}\leavevmode {\color {Chapter }GroupHomomorphismByImagesNC}}{590}{subsection.40.1.2}
\contentsline {subsection}{\numberline {40.1.3}\leavevmode {\color {Chapter }GroupGeneralMappingByImages}}{590}{subsection.40.1.3}
\contentsline {subsection}{\numberline {40.1.4}\leavevmode {\color {Chapter }GroupHomomorphismByFunction}}{591}{subsection.40.1.4}
\contentsline {subsection}{\numberline {40.1.5}\leavevmode {\color {Chapter }AsGroupGeneralMappingByImages}}{591}{subsection.40.1.5}
\contentsline {section}{\numberline {40.2}\leavevmode {\color {Chapter }Operations for Group Homomorphisms}}{592}{section.40.2}
\contentsline {section}{\numberline {40.3}\leavevmode {\color {Chapter }Efficiency of Homomorphisms}}{593}{section.40.3}
\contentsline {subsection}{\numberline {40.3.1}\leavevmode {\color {Chapter }Mappings given on generators}}{593}{subsection.40.3.1}
\contentsline {subsection}{\numberline {40.3.2}\leavevmode {\color {Chapter }Action homomorphisms}}{593}{subsection.40.3.2}
\contentsline {subsection}{\numberline {40.3.3}\leavevmode {\color {Chapter }Mappings given by functions}}{594}{subsection.40.3.3}
\contentsline {subsection}{\numberline {40.3.4}\leavevmode {\color {Chapter }Other operations}}{594}{subsection.40.3.4}
\contentsline {subsection}{\numberline {40.3.5}\leavevmode {\color {Chapter }ImagesSmallestGenerators}}{594}{subsection.40.3.5}
\contentsline {section}{\numberline {40.4}\leavevmode {\color {Chapter }Homomorphism for very large groups}}{594}{section.40.4}
\contentsline {section}{\numberline {40.5}\leavevmode {\color {Chapter }Nice Monomorphisms}}{595}{section.40.5}
\contentsline {subsection}{\numberline {40.5.1}\leavevmode {\color {Chapter }IsHandledByNiceMonomorphism}}{595}{subsection.40.5.1}
\contentsline {subsection}{\numberline {40.5.2}\leavevmode {\color {Chapter }NiceMonomorphism}}{595}{subsection.40.5.2}
\contentsline {subsection}{\numberline {40.5.3}\leavevmode {\color {Chapter }NiceObject}}{595}{subsection.40.5.3}
\contentsline {subsection}{\numberline {40.5.4}\leavevmode {\color {Chapter }IsCanonicalNiceMonomorphism}}{596}{subsection.40.5.4}
\contentsline {section}{\numberline {40.6}\leavevmode {\color {Chapter }Group Automorphisms}}{596}{section.40.6}
\contentsline {subsection}{\numberline {40.6.1}\leavevmode {\color {Chapter }ConjugatorIsomorphism}}{596}{subsection.40.6.1}
\contentsline {subsection}{\numberline {40.6.2}\leavevmode {\color {Chapter }ConjugatorAutomorphism}}{596}{subsection.40.6.2}
\contentsline {subsection}{\numberline {40.6.3}\leavevmode {\color {Chapter }InnerAutomorphism}}{596}{subsection.40.6.3}
\contentsline {subsection}{\numberline {40.6.4}\leavevmode {\color {Chapter }IsConjugatorIsomorphism}}{597}{subsection.40.6.4}
\contentsline {subsection}{\numberline {40.6.5}\leavevmode {\color {Chapter }ConjugatorOfConjugatorIsomorphism}}{597}{subsection.40.6.5}
\contentsline {section}{\numberline {40.7}\leavevmode {\color {Chapter }Groups of Automorphisms}}{598}{section.40.7}
\contentsline {subsection}{\numberline {40.7.1}\leavevmode {\color {Chapter }AutomorphismGroup}}{598}{subsection.40.7.1}
\contentsline {subsection}{\numberline {40.7.2}\leavevmode {\color {Chapter }IsGroupOfAutomorphisms}}{598}{subsection.40.7.2}
\contentsline {subsection}{\numberline {40.7.3}\leavevmode {\color {Chapter }AutomorphismDomain}}{598}{subsection.40.7.3}
\contentsline {subsection}{\numberline {40.7.4}\leavevmode {\color {Chapter }IsAutomorphismGroup}}{598}{subsection.40.7.4}
\contentsline {subsection}{\numberline {40.7.5}\leavevmode {\color {Chapter }InnerAutomorphismsAutomorphismGroup}}{599}{subsection.40.7.5}
\contentsline {subsection}{\numberline {40.7.6}\leavevmode {\color {Chapter }InducedAutomorphism}}{599}{subsection.40.7.6}
\contentsline {section}{\numberline {40.8}\leavevmode {\color {Chapter }Calculating with Group Automorphisms}}{599}{section.40.8}
\contentsline {subsection}{\numberline {40.8.1}\leavevmode {\color {Chapter }AssignNiceMonomorphismAutomorphismGroup}}{600}{subsection.40.8.1}
\contentsline {subsection}{\numberline {40.8.2}\leavevmode {\color {Chapter }NiceMonomorphismAutomGroup}}{600}{subsection.40.8.2}
\contentsline {section}{\numberline {40.9}\leavevmode {\color {Chapter }Searching for Homomorphisms}}{600}{section.40.9}
\contentsline {subsection}{\numberline {40.9.1}\leavevmode {\color {Chapter }IsomorphismGroups}}{600}{subsection.40.9.1}
\contentsline {subsection}{\numberline {40.9.2}\leavevmode {\color {Chapter }AllHomomorphismClasses}}{601}{subsection.40.9.2}
\contentsline {subsection}{\numberline {40.9.3}\leavevmode {\color {Chapter }AllHomomorphisms}}{601}{subsection.40.9.3}
\contentsline {subsection}{\numberline {40.9.4}\leavevmode {\color {Chapter }GQuotients}}{601}{subsection.40.9.4}
\contentsline {subsection}{\numberline {40.9.5}\leavevmode {\color {Chapter }IsomorphicSubgroups}}{602}{subsection.40.9.5}
\contentsline {subsection}{\numberline {40.9.6}\leavevmode {\color {Chapter }MorClassLoop}}{602}{subsection.40.9.6}
\contentsline {section}{\numberline {40.10}\leavevmode {\color {Chapter }Representations for Group Homomorphisms}}{603}{section.40.10}
\contentsline {subsection}{\numberline {40.10.1}\leavevmode {\color {Chapter }IsGroupGeneralMappingByImages}}{604}{subsection.40.10.1}
\contentsline {subsection}{\numberline {40.10.2}\leavevmode {\color {Chapter }MappingGeneratorsImages}}{604}{subsection.40.10.2}
\contentsline {subsection}{\numberline {40.10.3}\leavevmode {\color {Chapter }IsGroupGeneralMappingByAsGroupGeneralMappingByImages}}{604}{subsection.40.10.3}
\contentsline {subsection}{\numberline {40.10.4}\leavevmode {\color {Chapter }IsPreimagesByAsGroupGeneralMappingByImages}}{604}{subsection.40.10.4}
\contentsline {subsection}{\numberline {40.10.5}\leavevmode {\color {Chapter }IsPermGroupGeneralMapping}}{604}{subsection.40.10.5}
\contentsline {subsection}{\numberline {40.10.6}\leavevmode {\color {Chapter }IsToPermGroupGeneralMappingByImages}}{604}{subsection.40.10.6}
\contentsline {subsection}{\numberline {40.10.7}\leavevmode {\color {Chapter }IsGroupGeneralMappingByPcgs}}{605}{subsection.40.10.7}
\contentsline {subsection}{\numberline {40.10.8}\leavevmode {\color {Chapter }IsPcGroupGeneralMappingByImages}}{605}{subsection.40.10.8}
\contentsline {subsection}{\numberline {40.10.9}\leavevmode {\color {Chapter }IsToPcGroupGeneralMappingByImages}}{605}{subsection.40.10.9}
\contentsline {subsection}{\numberline {40.10.10}\leavevmode {\color {Chapter }IsFromFpGroupGeneralMappingByImages}}{605}{subsection.40.10.10}
\contentsline {subsection}{\numberline {40.10.11}\leavevmode {\color {Chapter }IsFromFpGroupStdGensGeneralMappingByImages}}{605}{subsection.40.10.11}
\contentsline {chapter}{\numberline {41}\leavevmode {\color {Chapter }Group Actions}}{606}{chapter.41}
\contentsline {section}{\numberline {41.1}\leavevmode {\color {Chapter }About Group Actions}}{606}{section.41.1}
\contentsline {section}{\numberline {41.2}\leavevmode {\color {Chapter }Basic Actions}}{607}{section.41.2}
\contentsline {subsection}{\numberline {41.2.1}\leavevmode {\color {Chapter }OnPoints}}{607}{subsection.41.2.1}
\contentsline {subsection}{\numberline {41.2.2}\leavevmode {\color {Chapter }OnRight}}{607}{subsection.41.2.2}
\contentsline {subsection}{\numberline {41.2.3}\leavevmode {\color {Chapter }OnLeftInverse}}{608}{subsection.41.2.3}
\contentsline {subsection}{\numberline {41.2.4}\leavevmode {\color {Chapter }OnSets}}{608}{subsection.41.2.4}
\contentsline {subsection}{\numberline {41.2.5}\leavevmode {\color {Chapter }OnTuples}}{608}{subsection.41.2.5}
\contentsline {subsection}{\numberline {41.2.6}\leavevmode {\color {Chapter }OnPairs}}{608}{subsection.41.2.6}
\contentsline {subsection}{\numberline {41.2.7}\leavevmode {\color {Chapter }OnSetsSets}}{608}{subsection.41.2.7}
\contentsline {subsection}{\numberline {41.2.8}\leavevmode {\color {Chapter }OnSetsDisjointSets}}{609}{subsection.41.2.8}
\contentsline {subsection}{\numberline {41.2.9}\leavevmode {\color {Chapter }OnSetsTuples}}{609}{subsection.41.2.9}
\contentsline {subsection}{\numberline {41.2.10}\leavevmode {\color {Chapter }OnTuplesSets}}{609}{subsection.41.2.10}
\contentsline {subsection}{\numberline {41.2.11}\leavevmode {\color {Chapter }OnTuplesTuples}}{609}{subsection.41.2.11}
\contentsline {subsection}{\numberline {41.2.12}\leavevmode {\color {Chapter }OnLines}}{610}{subsection.41.2.12}
\contentsline {subsection}{\numberline {41.2.13}\leavevmode {\color {Chapter }OnIndeterminates (as a permutation action)}}{610}{subsection.41.2.13}
\contentsline {subsection}{\numberline {41.2.14}\leavevmode {\color {Chapter }Permuted (as a permutation action)}}{610}{subsection.41.2.14}
\contentsline {subsection}{\numberline {41.2.15}\leavevmode {\color {Chapter }OnSubspacesByCanonicalBasis}}{611}{subsection.41.2.15}
\contentsline {section}{\numberline {41.3}\leavevmode {\color {Chapter }Action on canonical representatives}}{611}{section.41.3}
\contentsline {section}{\numberline {41.4}\leavevmode {\color {Chapter }Orbits}}{611}{section.41.4}
\contentsline {subsection}{\numberline {41.4.1}\leavevmode {\color {Chapter }Orbit}}{611}{subsection.41.4.1}
\contentsline {subsection}{\numberline {41.4.2}\leavevmode {\color {Chapter }Orbits (operation)}}{612}{subsection.41.4.2}
\contentsline {subsection}{\numberline {41.4.3}\leavevmode {\color {Chapter }OrbitsDomain}}{612}{subsection.41.4.3}
\contentsline {subsection}{\numberline {41.4.4}\leavevmode {\color {Chapter }OrbitLength}}{613}{subsection.41.4.4}
\contentsline {subsection}{\numberline {41.4.5}\leavevmode {\color {Chapter }OrbitLengths}}{613}{subsection.41.4.5}
\contentsline {subsection}{\numberline {41.4.6}\leavevmode {\color {Chapter }OrbitLengthsDomain}}{613}{subsection.41.4.6}
\contentsline {section}{\numberline {41.5}\leavevmode {\color {Chapter }Stabilizers}}{614}{section.41.5}
\contentsline {subsection}{\numberline {41.5.1}\leavevmode {\color {Chapter }OrbitStabilizer}}{614}{subsection.41.5.1}
\contentsline {subsection}{\numberline {41.5.2}\leavevmode {\color {Chapter }Stabilizer}}{614}{subsection.41.5.2}
\contentsline {subsection}{\numberline {41.5.3}\leavevmode {\color {Chapter }OrbitStabilizerAlgorithm}}{614}{subsection.41.5.3}
\contentsline {section}{\numberline {41.6}\leavevmode {\color {Chapter }Elements with Prescribed Images}}{615}{section.41.6}
\contentsline {subsection}{\numberline {41.6.1}\leavevmode {\color {Chapter }RepresentativeAction}}{615}{subsection.41.6.1}
\contentsline {section}{\numberline {41.7}\leavevmode {\color {Chapter }The Permutation Image of an Action}}{616}{section.41.7}
\contentsline {subsection}{\numberline {41.7.1}\leavevmode {\color {Chapter }ActionHomomorphism}}{616}{subsection.41.7.1}
\contentsline {subsection}{\numberline {41.7.2}\leavevmode {\color {Chapter }Action (for a group, an action domain, etc.)}}{616}{subsection.41.7.2}
\contentsline {subsection}{\numberline {41.7.3}\leavevmode {\color {Chapter }SparseActionHomomorphism}}{617}{subsection.41.7.3}
\contentsline {section}{\numberline {41.8}\leavevmode {\color {Chapter }Action of a group on itself}}{617}{section.41.8}
\contentsline {subsection}{\numberline {41.8.1}\leavevmode {\color {Chapter }FactorCosetAction}}{618}{subsection.41.8.1}
\contentsline {subsection}{\numberline {41.8.2}\leavevmode {\color {Chapter }RegularActionHomomorphism}}{618}{subsection.41.8.2}
\contentsline {subsection}{\numberline {41.8.3}\leavevmode {\color {Chapter }AbelianSubfactorAction}}{618}{subsection.41.8.3}
\contentsline {section}{\numberline {41.9}\leavevmode {\color {Chapter }Permutations Induced by Elements and Cycles}}{619}{section.41.9}
\contentsline {subsection}{\numberline {41.9.1}\leavevmode {\color {Chapter }Permutation}}{619}{subsection.41.9.1}
\contentsline {subsection}{\numberline {41.9.2}\leavevmode {\color {Chapter }PermutationCycle}}{619}{subsection.41.9.2}
\contentsline {subsection}{\numberline {41.9.3}\leavevmode {\color {Chapter }Cycle}}{619}{subsection.41.9.3}
\contentsline {subsection}{\numberline {41.9.4}\leavevmode {\color {Chapter }CycleLength}}{619}{subsection.41.9.4}
\contentsline {subsection}{\numberline {41.9.5}\leavevmode {\color {Chapter }Cycles}}{620}{subsection.41.9.5}
\contentsline {subsection}{\numberline {41.9.6}\leavevmode {\color {Chapter }CycleLengths}}{620}{subsection.41.9.6}
\contentsline {subsection}{\numberline {41.9.7}\leavevmode {\color {Chapter }CycleIndex}}{620}{subsection.41.9.7}
\contentsline {section}{\numberline {41.10}\leavevmode {\color {Chapter }Tests for Actions}}{620}{section.41.10}
\contentsline {subsection}{\numberline {41.10.1}\leavevmode {\color {Chapter }IsTransitive}}{620}{subsection.41.10.1}
\contentsline {subsection}{\numberline {41.10.2}\leavevmode {\color {Chapter }Transitivity}}{621}{subsection.41.10.2}
\contentsline {subsection}{\numberline {41.10.3}\leavevmode {\color {Chapter }RankAction}}{621}{subsection.41.10.3}
\contentsline {subsection}{\numberline {41.10.4}\leavevmode {\color {Chapter }IsSemiRegular}}{621}{subsection.41.10.4}
\contentsline {subsection}{\numberline {41.10.5}\leavevmode {\color {Chapter }IsRegular}}{622}{subsection.41.10.5}
\contentsline {subsection}{\numberline {41.10.6}\leavevmode {\color {Chapter }Earns}}{622}{subsection.41.10.6}
\contentsline {subsection}{\numberline {41.10.7}\leavevmode {\color {Chapter }IsPrimitive}}{622}{subsection.41.10.7}
\contentsline {section}{\numberline {41.11}\leavevmode {\color {Chapter }Block Systems}}{623}{section.41.11}
\contentsline {subsection}{\numberline {41.11.1}\leavevmode {\color {Chapter }Blocks}}{623}{subsection.41.11.1}
\contentsline {subsection}{\numberline {41.11.2}\leavevmode {\color {Chapter }MaximalBlocks}}{623}{subsection.41.11.2}
\contentsline {subsection}{\numberline {41.11.3}\leavevmode {\color {Chapter }RepresentativesMinimalBlocks}}{623}{subsection.41.11.3}
\contentsline {subsection}{\numberline {41.11.4}\leavevmode {\color {Chapter }AllBlocks}}{624}{subsection.41.11.4}
\contentsline {section}{\numberline {41.12}\leavevmode {\color {Chapter }External Sets}}{624}{section.41.12}
\contentsline {subsection}{\numberline {41.12.1}\leavevmode {\color {Chapter }IsExternalSet}}{625}{subsection.41.12.1}
\contentsline {subsection}{\numberline {41.12.2}\leavevmode {\color {Chapter }ExternalSet}}{625}{subsection.41.12.2}
\contentsline {subsection}{\numberline {41.12.3}\leavevmode {\color {Chapter }ActingDomain}}{625}{subsection.41.12.3}
\contentsline {subsection}{\numberline {41.12.4}\leavevmode {\color {Chapter }FunctionAction}}{625}{subsection.41.12.4}
\contentsline {subsection}{\numberline {41.12.5}\leavevmode {\color {Chapter }HomeEnumerator}}{625}{subsection.41.12.5}
\contentsline {subsection}{\numberline {41.12.6}\leavevmode {\color {Chapter }IsExternalSubset}}{626}{subsection.41.12.6}
\contentsline {subsection}{\numberline {41.12.7}\leavevmode {\color {Chapter }ExternalSubset}}{626}{subsection.41.12.7}
\contentsline {subsection}{\numberline {41.12.8}\leavevmode {\color {Chapter }IsExternalOrbit}}{626}{subsection.41.12.8}
\contentsline {subsection}{\numberline {41.12.9}\leavevmode {\color {Chapter }ExternalOrbit}}{626}{subsection.41.12.9}
\contentsline {subsection}{\numberline {41.12.10}\leavevmode {\color {Chapter }StabilizerOfExternalSet}}{626}{subsection.41.12.10}
\contentsline {subsection}{\numberline {41.12.11}\leavevmode {\color {Chapter }ExternalOrbits}}{627}{subsection.41.12.11}
\contentsline {subsection}{\numberline {41.12.12}\leavevmode {\color {Chapter }ExternalOrbitsStabilizers}}{627}{subsection.41.12.12}
\contentsline {subsection}{\numberline {41.12.13}\leavevmode {\color {Chapter }CanonicalRepresentativeOfExternalSet}}{627}{subsection.41.12.13}
\contentsline {subsection}{\numberline {41.12.14}\leavevmode {\color {Chapter }CanonicalRepresentativeDeterminatorOfExternalSet}}{627}{subsection.41.12.14}
\contentsline {subsection}{\numberline {41.12.15}\leavevmode {\color {Chapter }ActorOfExternalSet}}{628}{subsection.41.12.15}
\contentsline {subsection}{\numberline {41.12.16}\leavevmode {\color {Chapter }UnderlyingExternalSet}}{628}{subsection.41.12.16}
\contentsline {subsection}{\numberline {41.12.17}\leavevmode {\color {Chapter }SurjectiveActionHomomorphismAttr}}{628}{subsection.41.12.17}
\contentsline {chapter}{\numberline {42}\leavevmode {\color {Chapter }Permutations}}{629}{chapter.42}
\contentsline {section}{\numberline {42.1}\leavevmode {\color {Chapter }IsPerm (Filter)}}{629}{section.42.1}
\contentsline {subsection}{\numberline {42.1.1}\leavevmode {\color {Chapter }IsPerm}}{630}{subsection.42.1.1}
\contentsline {subsection}{\numberline {42.1.2}\leavevmode {\color {Chapter }IsPermCollection}}{630}{subsection.42.1.2}
\contentsline {subsection}{\numberline {42.1.3}\leavevmode {\color {Chapter }PermutationsFamily}}{630}{subsection.42.1.3}
\contentsline {subsection}{\numberline {42.1.4}\leavevmode {\color {Chapter }PERM{\textunderscore }INVERSE{\textunderscore }THRESHOLD}}{630}{subsection.42.1.4}
\contentsline {section}{\numberline {42.2}\leavevmode {\color {Chapter }Comparison of Permutations}}{631}{section.42.2}
\contentsline {subsection}{\numberline {42.2.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= (for permutations)}}{631}{subsection.42.2.1}
\contentsline {subsection}{\numberline {42.2.2}\leavevmode {\color {Chapter }DistancePerms}}{631}{subsection.42.2.2}
\contentsline {subsection}{\numberline {42.2.3}\leavevmode {\color {Chapter }SmallestGeneratorPerm}}{631}{subsection.42.2.3}
\contentsline {section}{\numberline {42.3}\leavevmode {\color {Chapter }Moved Points of Permutations}}{631}{section.42.3}
\contentsline {subsection}{\numberline {42.3.1}\leavevmode {\color {Chapter }SmallestMovedPoint (for a permutation)}}{631}{subsection.42.3.1}
\contentsline {subsection}{\numberline {42.3.2}\leavevmode {\color {Chapter }LargestMovedPoint (for a permutation)}}{632}{subsection.42.3.2}
\contentsline {subsection}{\numberline {42.3.3}\leavevmode {\color {Chapter }MovedPoints (for a permutation)}}{632}{subsection.42.3.3}
\contentsline {subsection}{\numberline {42.3.4}\leavevmode {\color {Chapter }NrMovedPoints (for a permutation)}}{632}{subsection.42.3.4}
\contentsline {section}{\numberline {42.4}\leavevmode {\color {Chapter }Sign and Cycle Structure}}{633}{section.42.4}
\contentsline {subsection}{\numberline {42.4.1}\leavevmode {\color {Chapter }SignPerm}}{633}{subsection.42.4.1}
\contentsline {subsection}{\numberline {42.4.2}\leavevmode {\color {Chapter }CycleStructurePerm}}{633}{subsection.42.4.2}
\contentsline {section}{\numberline {42.5}\leavevmode {\color {Chapter }Creating Permutations}}{633}{section.42.5}
\contentsline {subsection}{\numberline {42.5.1}\leavevmode {\color {Chapter }ListPerm}}{633}{subsection.42.5.1}
\contentsline {subsection}{\numberline {42.5.2}\leavevmode {\color {Chapter }PermList}}{633}{subsection.42.5.2}
\contentsline {subsection}{\numberline {42.5.3}\leavevmode {\color {Chapter }MappingPermListList}}{634}{subsection.42.5.3}
\contentsline {subsection}{\numberline {42.5.4}\leavevmode {\color {Chapter }RestrictedPerm}}{634}{subsection.42.5.4}
\contentsline {subsection}{\numberline {42.5.5}\leavevmode {\color {Chapter }CycleFromList}}{634}{subsection.42.5.5}
\contentsline {subsection}{\numberline {42.5.6}\leavevmode {\color {Chapter }AsPermutation}}{635}{subsection.42.5.6}
\contentsline {chapter}{\numberline {43}\leavevmode {\color {Chapter }Permutation Groups}}{636}{chapter.43}
\contentsline {section}{\numberline {43.1}\leavevmode {\color {Chapter }IsPermGroup (Filter)}}{636}{section.43.1}
\contentsline {subsection}{\numberline {43.1.1}\leavevmode {\color {Chapter }IsPermGroup}}{636}{subsection.43.1.1}
\contentsline {section}{\numberline {43.2}\leavevmode {\color {Chapter }The Natural Action}}{636}{section.43.2}
\contentsline {subsection}{\numberline {43.2.1}\leavevmode {\color {Chapter }OrbitPerms}}{637}{subsection.43.2.1}
\contentsline {subsection}{\numberline {43.2.2}\leavevmode {\color {Chapter }OrbitsPerms}}{637}{subsection.43.2.2}
\contentsline {section}{\numberline {43.3}\leavevmode {\color {Chapter }Computing a Permutation Representation}}{637}{section.43.3}
\contentsline {subsection}{\numberline {43.3.1}\leavevmode {\color {Chapter }IsomorphismPermGroup}}{637}{subsection.43.3.1}
\contentsline {subsection}{\numberline {43.3.2}\leavevmode {\color {Chapter }SmallerDegreePermutationRepresentation}}{638}{subsection.43.3.2}
\contentsline {section}{\numberline {43.4}\leavevmode {\color {Chapter }Symmetric and Alternating Groups}}{638}{section.43.4}
\contentsline {subsection}{\numberline {43.4.1}\leavevmode {\color {Chapter }IsNaturalSymmetricGroup}}{639}{subsection.43.4.1}
\contentsline {subsection}{\numberline {43.4.2}\leavevmode {\color {Chapter }IsSymmetricGroup}}{639}{subsection.43.4.2}
\contentsline {subsection}{\numberline {43.4.3}\leavevmode {\color {Chapter }IsAlternatingGroup}}{639}{subsection.43.4.3}
\contentsline {subsection}{\numberline {43.4.4}\leavevmode {\color {Chapter }SymmetricParentGroup}}{639}{subsection.43.4.4}
\contentsline {section}{\numberline {43.5}\leavevmode {\color {Chapter }Primitive Groups}}{639}{section.43.5}
\contentsline {subsection}{\numberline {43.5.1}\leavevmode {\color {Chapter }ONanScottType}}{639}{subsection.43.5.1}
\contentsline {subsection}{\numberline {43.5.2}\leavevmode {\color {Chapter }SocleTypePrimitiveGroup}}{640}{subsection.43.5.2}
\contentsline {section}{\numberline {43.6}\leavevmode {\color {Chapter }Stabilizer Chains}}{641}{section.43.6}
\contentsline {section}{\numberline {43.7}\leavevmode {\color {Chapter }Randomized Methods for Permutation Groups}}{641}{section.43.7}
\contentsline {section}{\numberline {43.8}\leavevmode {\color {Chapter }Construction of Stabilizer Chains}}{644}{section.43.8}
\contentsline {subsection}{\numberline {43.8.1}\leavevmode {\color {Chapter }StabChain (for a group (and a record))}}{644}{subsection.43.8.1}
\contentsline {subsection}{\numberline {43.8.2}\leavevmode {\color {Chapter }StabChainOptions}}{645}{subsection.43.8.2}
\contentsline {subsection}{\numberline {43.8.3}\leavevmode {\color {Chapter }DefaultStabChainOptions}}{645}{subsection.43.8.3}
\contentsline {subsection}{\numberline {43.8.4}\leavevmode {\color {Chapter }StabChainBaseStrongGenerators}}{645}{subsection.43.8.4}
\contentsline {subsection}{\numberline {43.8.5}\leavevmode {\color {Chapter }MinimalStabChain}}{646}{subsection.43.8.5}
\contentsline {section}{\numberline {43.9}\leavevmode {\color {Chapter }Stabilizer Chain Records}}{646}{section.43.9}
\contentsline {section}{\numberline {43.10}\leavevmode {\color {Chapter }Operations for Stabilizer Chains}}{647}{section.43.10}
\contentsline {subsection}{\numberline {43.10.1}\leavevmode {\color {Chapter }BaseStabChain}}{647}{subsection.43.10.1}
\contentsline {subsection}{\numberline {43.10.2}\leavevmode {\color {Chapter }BaseOfGroup}}{647}{subsection.43.10.2}
\contentsline {subsection}{\numberline {43.10.3}\leavevmode {\color {Chapter }SizeStabChain}}{648}{subsection.43.10.3}
\contentsline {subsection}{\numberline {43.10.4}\leavevmode {\color {Chapter }StrongGeneratorsStabChain}}{648}{subsection.43.10.4}
\contentsline {subsection}{\numberline {43.10.5}\leavevmode {\color {Chapter }GroupStabChain}}{648}{subsection.43.10.5}
\contentsline {subsection}{\numberline {43.10.6}\leavevmode {\color {Chapter }OrbitStabChain}}{648}{subsection.43.10.6}
\contentsline {subsection}{\numberline {43.10.7}\leavevmode {\color {Chapter }IndicesStabChain}}{648}{subsection.43.10.7}
\contentsline {subsection}{\numberline {43.10.8}\leavevmode {\color {Chapter }ListStabChain}}{648}{subsection.43.10.8}
\contentsline {subsection}{\numberline {43.10.9}\leavevmode {\color {Chapter }ElementsStabChain}}{648}{subsection.43.10.9}
\contentsline {subsection}{\numberline {43.10.10}\leavevmode {\color {Chapter }IteratorStabChain}}{649}{subsection.43.10.10}
\contentsline {subsection}{\numberline {43.10.11}\leavevmode {\color {Chapter }InverseRepresentative}}{649}{subsection.43.10.11}
\contentsline {subsection}{\numberline {43.10.12}\leavevmode {\color {Chapter }SiftedPermutation}}{649}{subsection.43.10.12}
\contentsline {subsection}{\numberline {43.10.13}\leavevmode {\color {Chapter }MinimalElementCosetStabChain}}{649}{subsection.43.10.13}
\contentsline {subsection}{\numberline {43.10.14}\leavevmode {\color {Chapter }LargestElementStabChain}}{649}{subsection.43.10.14}
\contentsline {subsection}{\numberline {43.10.15}\leavevmode {\color {Chapter }ApproximateSuborbitsStabilizerPermGroup}}{649}{subsection.43.10.15}
\contentsline {section}{\numberline {43.11}\leavevmode {\color {Chapter }Low Level Routines to Modify and Create Stabilizer Chains}}{650}{section.43.11}
\contentsline {subsection}{\numberline {43.11.1}\leavevmode {\color {Chapter }CopyStabChain}}{650}{subsection.43.11.1}
\contentsline {subsection}{\numberline {43.11.2}\leavevmode {\color {Chapter }CopyOptionsDefaults}}{650}{subsection.43.11.2}
\contentsline {subsection}{\numberline {43.11.3}\leavevmode {\color {Chapter }ChangeStabChain}}{650}{subsection.43.11.3}
\contentsline {subsection}{\numberline {43.11.4}\leavevmode {\color {Chapter }ExtendStabChain}}{650}{subsection.43.11.4}
\contentsline {subsection}{\numberline {43.11.5}\leavevmode {\color {Chapter }ReduceStabChain}}{650}{subsection.43.11.5}
\contentsline {subsection}{\numberline {43.11.6}\leavevmode {\color {Chapter }RemoveStabChain}}{651}{subsection.43.11.6}
\contentsline {subsection}{\numberline {43.11.7}\leavevmode {\color {Chapter }EmptyStabChain}}{651}{subsection.43.11.7}
\contentsline {subsection}{\numberline {43.11.8}\leavevmode {\color {Chapter }InsertTrivialStabilizer}}{651}{subsection.43.11.8}
\contentsline {subsection}{\numberline {43.11.9}\leavevmode {\color {Chapter }IsFixedStabilizer}}{651}{subsection.43.11.9}
\contentsline {subsection}{\numberline {43.11.10}\leavevmode {\color {Chapter }AddGeneratorsExtendSchreierTree}}{651}{subsection.43.11.10}
\contentsline {section}{\numberline {43.12}\leavevmode {\color {Chapter }Backtrack}}{651}{section.43.12}
\contentsline {subsection}{\numberline {43.12.1}\leavevmode {\color {Chapter }SubgroupProperty}}{652}{subsection.43.12.1}
\contentsline {subsection}{\numberline {43.12.2}\leavevmode {\color {Chapter }ElementProperty}}{652}{subsection.43.12.2}
\contentsline {subsection}{\numberline {43.12.3}\leavevmode {\color {Chapter }TwoClosure}}{652}{subsection.43.12.3}
\contentsline {subsection}{\numberline {43.12.4}\leavevmode {\color {Chapter }InfoBckt}}{653}{subsection.43.12.4}
\contentsline {section}{\numberline {43.13}\leavevmode {\color {Chapter }Working with large degree permutation groups}}{653}{section.43.13}
\contentsline {chapter}{\numberline {44}\leavevmode {\color {Chapter }Matrix Groups}}{655}{chapter.44}
\contentsline {section}{\numberline {44.1}\leavevmode {\color {Chapter }IsMatrixGroup (Filter)}}{655}{section.44.1}
\contentsline {subsection}{\numberline {44.1.1}\leavevmode {\color {Chapter }IsMatrixGroup}}{655}{subsection.44.1.1}
\contentsline {section}{\numberline {44.2}\leavevmode {\color {Chapter }Attributes and Properties for Matrix Groups}}{656}{section.44.2}
\contentsline {subsection}{\numberline {44.2.1}\leavevmode {\color {Chapter }DimensionOfMatrixGroup}}{656}{subsection.44.2.1}
\contentsline {subsection}{\numberline {44.2.2}\leavevmode {\color {Chapter }DefaultFieldOfMatrixGroup}}{656}{subsection.44.2.2}
\contentsline {subsection}{\numberline {44.2.3}\leavevmode {\color {Chapter }FieldOfMatrixGroup}}{656}{subsection.44.2.3}
\contentsline {subsection}{\numberline {44.2.4}\leavevmode {\color {Chapter }TransposedMatrixGroup}}{656}{subsection.44.2.4}
\contentsline {subsection}{\numberline {44.2.5}\leavevmode {\color {Chapter }IsFFEMatrixGroup}}{656}{subsection.44.2.5}
\contentsline {section}{\numberline {44.3}\leavevmode {\color {Chapter }Actions of Matrix Groups}}{657}{section.44.3}
\contentsline {subsection}{\numberline {44.3.1}\leavevmode {\color {Chapter }ProjectiveActionOnFullSpace}}{657}{subsection.44.3.1}
\contentsline {subsection}{\numberline {44.3.2}\leavevmode {\color {Chapter }ProjectiveActionHomomorphismMatrixGroup}}{657}{subsection.44.3.2}
\contentsline {subsection}{\numberline {44.3.3}\leavevmode {\color {Chapter }BlowUpIsomorphism}}{657}{subsection.44.3.3}
\contentsline {section}{\numberline {44.4}\leavevmode {\color {Chapter }GL and SL}}{657}{section.44.4}
\contentsline {subsection}{\numberline {44.4.1}\leavevmode {\color {Chapter }IsGeneralLinearGroup}}{658}{subsection.44.4.1}
\contentsline {subsection}{\numberline {44.4.2}\leavevmode {\color {Chapter }IsNaturalGL}}{658}{subsection.44.4.2}
\contentsline {subsection}{\numberline {44.4.3}\leavevmode {\color {Chapter }IsSpecialLinearGroup}}{658}{subsection.44.4.3}
\contentsline {subsection}{\numberline {44.4.4}\leavevmode {\color {Chapter }IsNaturalSL}}{658}{subsection.44.4.4}
\contentsline {subsection}{\numberline {44.4.5}\leavevmode {\color {Chapter }IsSubgroupSL}}{658}{subsection.44.4.5}
\contentsline {section}{\numberline {44.5}\leavevmode {\color {Chapter }Invariant Forms}}{659}{section.44.5}
\contentsline {subsection}{\numberline {44.5.1}\leavevmode {\color {Chapter }InvariantBilinearForm}}{659}{subsection.44.5.1}
\contentsline {subsection}{\numberline {44.5.2}\leavevmode {\color {Chapter }IsFullSubgroupGLorSLRespectingBilinearForm}}{659}{subsection.44.5.2}
\contentsline {subsection}{\numberline {44.5.3}\leavevmode {\color {Chapter }InvariantSesquilinearForm}}{659}{subsection.44.5.3}
\contentsline {subsection}{\numberline {44.5.4}\leavevmode {\color {Chapter }IsFullSubgroupGLorSLRespectingSesquilinearForm}}{659}{subsection.44.5.4}
\contentsline {subsection}{\numberline {44.5.5}\leavevmode {\color {Chapter }InvariantQuadraticForm}}{659}{subsection.44.5.5}
\contentsline {subsection}{\numberline {44.5.6}\leavevmode {\color {Chapter }IsFullSubgroupGLorSLRespectingQuadraticForm}}{660}{subsection.44.5.6}
\contentsline {section}{\numberline {44.6}\leavevmode {\color {Chapter }Matrix Groups in Characteristic 0}}{660}{section.44.6}
\contentsline {subsection}{\numberline {44.6.1}\leavevmode {\color {Chapter }IsCyclotomicMatrixGroup}}{661}{subsection.44.6.1}
\contentsline {subsection}{\numberline {44.6.2}\leavevmode {\color {Chapter }IsRationalMatrixGroup}}{661}{subsection.44.6.2}
\contentsline {subsection}{\numberline {44.6.3}\leavevmode {\color {Chapter }IsIntegerMatrixGroup}}{661}{subsection.44.6.3}
\contentsline {subsection}{\numberline {44.6.4}\leavevmode {\color {Chapter }IsNaturalGLnZ}}{661}{subsection.44.6.4}
\contentsline {subsection}{\numberline {44.6.5}\leavevmode {\color {Chapter }IsNaturalSLnZ}}{661}{subsection.44.6.5}
\contentsline {subsection}{\numberline {44.6.6}\leavevmode {\color {Chapter }InvariantLattice}}{661}{subsection.44.6.6}
\contentsline {subsection}{\numberline {44.6.7}\leavevmode {\color {Chapter }NormalizerInGLnZ}}{662}{subsection.44.6.7}
\contentsline {subsection}{\numberline {44.6.8}\leavevmode {\color {Chapter }CentralizerInGLnZ}}{662}{subsection.44.6.8}
\contentsline {subsection}{\numberline {44.6.9}\leavevmode {\color {Chapter }ZClassRepsQClass}}{662}{subsection.44.6.9}
\contentsline {subsection}{\numberline {44.6.10}\leavevmode {\color {Chapter }IsBravaisGroup}}{662}{subsection.44.6.10}
\contentsline {subsection}{\numberline {44.6.11}\leavevmode {\color {Chapter }BravaisGroup}}{662}{subsection.44.6.11}
\contentsline {subsection}{\numberline {44.6.12}\leavevmode {\color {Chapter }BravaisSubgroups}}{662}{subsection.44.6.12}
\contentsline {subsection}{\numberline {44.6.13}\leavevmode {\color {Chapter }BravaisSupergroups}}{663}{subsection.44.6.13}
\contentsline {subsection}{\numberline {44.6.14}\leavevmode {\color {Chapter }NormalizerInGLnZBravaisGroup}}{663}{subsection.44.6.14}
\contentsline {section}{\numberline {44.7}\leavevmode {\color {Chapter }Acting OnRight and OnLeft}}{663}{section.44.7}
\contentsline {subsection}{\numberline {44.7.1}\leavevmode {\color {Chapter }CrystGroupDefaultAction}}{663}{subsection.44.7.1}
\contentsline {subsection}{\numberline {44.7.2}\leavevmode {\color {Chapter }SetCrystGroupDefaultAction}}{663}{subsection.44.7.2}
\contentsline {chapter}{\numberline {45}\leavevmode {\color {Chapter }Polycyclic Groups}}{664}{chapter.45}
\contentsline {section}{\numberline {45.1}\leavevmode {\color {Chapter }Polycyclic Generating Systems}}{664}{section.45.1}
\contentsline {section}{\numberline {45.2}\leavevmode {\color {Chapter }Computing a Pcgs}}{665}{section.45.2}
\contentsline {subsection}{\numberline {45.2.1}\leavevmode {\color {Chapter }Pcgs}}{665}{subsection.45.2.1}
\contentsline {subsection}{\numberline {45.2.2}\leavevmode {\color {Chapter }IsPcgs}}{665}{subsection.45.2.2}
\contentsline {subsection}{\numberline {45.2.3}\leavevmode {\color {Chapter }CanEasilyComputePcgs}}{665}{subsection.45.2.3}
\contentsline {section}{\numberline {45.3}\leavevmode {\color {Chapter }Defining a Pcgs Yourself}}{666}{section.45.3}
\contentsline {subsection}{\numberline {45.3.1}\leavevmode {\color {Chapter }PcgsByPcSequence}}{666}{subsection.45.3.1}
\contentsline {section}{\numberline {45.4}\leavevmode {\color {Chapter }Elementary Operations for a Pcgs}}{666}{section.45.4}
\contentsline {subsection}{\numberline {45.4.1}\leavevmode {\color {Chapter }RelativeOrders}}{666}{subsection.45.4.1}
\contentsline {subsection}{\numberline {45.4.2}\leavevmode {\color {Chapter }IsFiniteOrdersPcgs}}{666}{subsection.45.4.2}
\contentsline {subsection}{\numberline {45.4.3}\leavevmode {\color {Chapter }IsPrimeOrdersPcgs}}{666}{subsection.45.4.3}
\contentsline {subsection}{\numberline {45.4.4}\leavevmode {\color {Chapter }PcSeries}}{667}{subsection.45.4.4}
\contentsline {subsection}{\numberline {45.4.5}\leavevmode {\color {Chapter }GroupOfPcgs}}{667}{subsection.45.4.5}
\contentsline {subsection}{\numberline {45.4.6}\leavevmode {\color {Chapter }OneOfPcgs}}{667}{subsection.45.4.6}
\contentsline {section}{\numberline {45.5}\leavevmode {\color {Chapter }Elementary Operations for a Pcgs and an Element}}{667}{section.45.5}
\contentsline {subsection}{\numberline {45.5.1}\leavevmode {\color {Chapter }RelativeOrderOfPcElement}}{667}{subsection.45.5.1}
\contentsline {subsection}{\numberline {45.5.2}\leavevmode {\color {Chapter }ExponentOfPcElement}}{667}{subsection.45.5.2}
\contentsline {subsection}{\numberline {45.5.3}\leavevmode {\color {Chapter }ExponentsOfPcElement}}{667}{subsection.45.5.3}
\contentsline {subsection}{\numberline {45.5.4}\leavevmode {\color {Chapter }DepthOfPcElement}}{668}{subsection.45.5.4}
\contentsline {subsection}{\numberline {45.5.5}\leavevmode {\color {Chapter }LeadingExponentOfPcElement}}{668}{subsection.45.5.5}
\contentsline {subsection}{\numberline {45.5.6}\leavevmode {\color {Chapter }PcElementByExponents}}{668}{subsection.45.5.6}
\contentsline {subsection}{\numberline {45.5.7}\leavevmode {\color {Chapter }LinearCombinationPcgs}}{668}{subsection.45.5.7}
\contentsline {subsection}{\numberline {45.5.8}\leavevmode {\color {Chapter }SiftedPcElement}}{668}{subsection.45.5.8}
\contentsline {subsection}{\numberline {45.5.9}\leavevmode {\color {Chapter }CanonicalPcElement}}{669}{subsection.45.5.9}
\contentsline {subsection}{\numberline {45.5.10}\leavevmode {\color {Chapter }ReducedPcElement}}{669}{subsection.45.5.10}
\contentsline {subsection}{\numberline {45.5.11}\leavevmode {\color {Chapter }CleanedTailPcElement}}{669}{subsection.45.5.11}
\contentsline {subsection}{\numberline {45.5.12}\leavevmode {\color {Chapter }HeadPcElementByNumber}}{669}{subsection.45.5.12}
\contentsline {section}{\numberline {45.6}\leavevmode {\color {Chapter }Exponents of Special Products}}{669}{section.45.6}
\contentsline {subsection}{\numberline {45.6.1}\leavevmode {\color {Chapter }ExponentsConjugateLayer}}{670}{subsection.45.6.1}
\contentsline {subsection}{\numberline {45.6.2}\leavevmode {\color {Chapter }ExponentsOfRelativePower}}{670}{subsection.45.6.2}
\contentsline {subsection}{\numberline {45.6.3}\leavevmode {\color {Chapter }ExponentsOfConjugate}}{670}{subsection.45.6.3}
\contentsline {subsection}{\numberline {45.6.4}\leavevmode {\color {Chapter }ExponentsOfCommutator}}{670}{subsection.45.6.4}
\contentsline {section}{\numberline {45.7}\leavevmode {\color {Chapter }Subgroups of Polycyclic Groups {\textendash } Induced Pcgs}}{670}{section.45.7}
\contentsline {subsection}{\numberline {45.7.1}\leavevmode {\color {Chapter }IsInducedPcgs}}{671}{subsection.45.7.1}
\contentsline {subsection}{\numberline {45.7.2}\leavevmode {\color {Chapter }InducedPcgsByPcSequence}}{671}{subsection.45.7.2}
\contentsline {subsection}{\numberline {45.7.3}\leavevmode {\color {Chapter }ParentPcgs}}{671}{subsection.45.7.3}
\contentsline {subsection}{\numberline {45.7.4}\leavevmode {\color {Chapter }InducedPcgs}}{671}{subsection.45.7.4}
\contentsline {subsection}{\numberline {45.7.5}\leavevmode {\color {Chapter }InducedPcgsByGenerators}}{672}{subsection.45.7.5}
\contentsline {subsection}{\numberline {45.7.6}\leavevmode {\color {Chapter }InducedPcgsByPcSequenceAndGenerators}}{672}{subsection.45.7.6}
\contentsline {subsection}{\numberline {45.7.7}\leavevmode {\color {Chapter }LeadCoeffsIGS}}{672}{subsection.45.7.7}
\contentsline {subsection}{\numberline {45.7.8}\leavevmode {\color {Chapter }ExtendedPcgs}}{672}{subsection.45.7.8}
\contentsline {subsection}{\numberline {45.7.9}\leavevmode {\color {Chapter }SubgroupByPcgs}}{672}{subsection.45.7.9}
\contentsline {section}{\numberline {45.8}\leavevmode {\color {Chapter }Subgroups of Polycyclic Groups {\textendash } Canonical Pcgs}}{672}{section.45.8}
\contentsline {subsection}{\numberline {45.8.1}\leavevmode {\color {Chapter }IsCanonicalPcgs}}{673}{subsection.45.8.1}
\contentsline {subsection}{\numberline {45.8.2}\leavevmode {\color {Chapter }CanonicalPcgs}}{673}{subsection.45.8.2}
\contentsline {section}{\numberline {45.9}\leavevmode {\color {Chapter }Factor Groups of Polycyclic Groups {\textendash } Modulo Pcgs}}{673}{section.45.9}
\contentsline {subsection}{\numberline {45.9.1}\leavevmode {\color {Chapter }ModuloPcgs}}{673}{subsection.45.9.1}
\contentsline {subsection}{\numberline {45.9.2}\leavevmode {\color {Chapter }IsModuloPcgs}}{674}{subsection.45.9.2}
\contentsline {subsection}{\numberline {45.9.3}\leavevmode {\color {Chapter }NumeratorOfModuloPcgs}}{674}{subsection.45.9.3}
\contentsline {subsection}{\numberline {45.9.4}\leavevmode {\color {Chapter }DenominatorOfModuloPcgs}}{674}{subsection.45.9.4}
\contentsline {subsection}{\numberline {45.9.5}\leavevmode {\color {Chapter }\texttt {\char 92\relax }mod (for two pcgs)}}{674}{subsection.45.9.5}
\contentsline {subsection}{\numberline {45.9.6}\leavevmode {\color {Chapter }CorrespondingGeneratorsByModuloPcgs}}{675}{subsection.45.9.6}
\contentsline {subsection}{\numberline {45.9.7}\leavevmode {\color {Chapter }CanonicalPcgsByGeneratorsWithImages}}{675}{subsection.45.9.7}
\contentsline {section}{\numberline {45.10}\leavevmode {\color {Chapter }Factor Groups of Polycyclic Groups in their Own Representation}}{675}{section.45.10}
\contentsline {subsection}{\numberline {45.10.1}\leavevmode {\color {Chapter }ProjectedPcElement}}{675}{subsection.45.10.1}
\contentsline {subsection}{\numberline {45.10.2}\leavevmode {\color {Chapter }ProjectedInducedPcgs}}{675}{subsection.45.10.2}
\contentsline {subsection}{\numberline {45.10.3}\leavevmode {\color {Chapter }LiftedPcElement}}{676}{subsection.45.10.3}
\contentsline {subsection}{\numberline {45.10.4}\leavevmode {\color {Chapter }LiftedInducedPcgs}}{676}{subsection.45.10.4}
\contentsline {section}{\numberline {45.11}\leavevmode {\color {Chapter }Pcgs and Normal Series}}{676}{section.45.11}
\contentsline {subsection}{\numberline {45.11.1}\leavevmode {\color {Chapter }IsPcgsElementaryAbelianSeries}}{676}{subsection.45.11.1}
\contentsline {subsection}{\numberline {45.11.2}\leavevmode {\color {Chapter }PcgsElementaryAbelianSeries (for a group)}}{676}{subsection.45.11.2}
\contentsline {subsection}{\numberline {45.11.3}\leavevmode {\color {Chapter }IndicesEANormalSteps}}{676}{subsection.45.11.3}
\contentsline {subsection}{\numberline {45.11.4}\leavevmode {\color {Chapter }EANormalSeriesByPcgs}}{677}{subsection.45.11.4}
\contentsline {subsection}{\numberline {45.11.5}\leavevmode {\color {Chapter }IsPcgsCentralSeries}}{677}{subsection.45.11.5}
\contentsline {subsection}{\numberline {45.11.6}\leavevmode {\color {Chapter }PcgsCentralSeries}}{677}{subsection.45.11.6}
\contentsline {subsection}{\numberline {45.11.7}\leavevmode {\color {Chapter }IndicesCentralNormalSteps}}{677}{subsection.45.11.7}
\contentsline {subsection}{\numberline {45.11.8}\leavevmode {\color {Chapter }CentralNormalSeriesByPcgs}}{678}{subsection.45.11.8}
\contentsline {subsection}{\numberline {45.11.9}\leavevmode {\color {Chapter }IsPcgsPCentralSeriesPGroup}}{678}{subsection.45.11.9}
\contentsline {subsection}{\numberline {45.11.10}\leavevmode {\color {Chapter }PcgsPCentralSeriesPGroup}}{678}{subsection.45.11.10}
\contentsline {subsection}{\numberline {45.11.11}\leavevmode {\color {Chapter }IndicesPCentralNormalStepsPGroup}}{678}{subsection.45.11.11}
\contentsline {subsection}{\numberline {45.11.12}\leavevmode {\color {Chapter }PCentralNormalSeriesByPcgsPGroup}}{679}{subsection.45.11.12}
\contentsline {subsection}{\numberline {45.11.13}\leavevmode {\color {Chapter }IsPcgsChiefSeries}}{679}{subsection.45.11.13}
\contentsline {subsection}{\numberline {45.11.14}\leavevmode {\color {Chapter }PcgsChiefSeries}}{679}{subsection.45.11.14}
\contentsline {subsection}{\numberline {45.11.15}\leavevmode {\color {Chapter }IndicesChiefNormalSteps}}{679}{subsection.45.11.15}
\contentsline {subsection}{\numberline {45.11.16}\leavevmode {\color {Chapter }ChiefNormalSeriesByPcgs}}{679}{subsection.45.11.16}
\contentsline {subsection}{\numberline {45.11.17}\leavevmode {\color {Chapter }IndicesNormalSteps}}{680}{subsection.45.11.17}
\contentsline {subsection}{\numberline {45.11.18}\leavevmode {\color {Chapter }NormalSeriesByPcgs}}{680}{subsection.45.11.18}
\contentsline {section}{\numberline {45.12}\leavevmode {\color {Chapter }Sum and Intersection of Pcgs}}{680}{section.45.12}
\contentsline {subsection}{\numberline {45.12.1}\leavevmode {\color {Chapter }SumFactorizationFunctionPcgs}}{680}{subsection.45.12.1}
\contentsline {section}{\numberline {45.13}\leavevmode {\color {Chapter }Special Pcgs}}{681}{section.45.13}
\contentsline {subsection}{\numberline {45.13.1}\leavevmode {\color {Chapter }IsSpecialPcgs}}{682}{subsection.45.13.1}
\contentsline {subsection}{\numberline {45.13.2}\leavevmode {\color {Chapter }SpecialPcgs}}{682}{subsection.45.13.2}
\contentsline {subsection}{\numberline {45.13.3}\leavevmode {\color {Chapter }LGWeights}}{682}{subsection.45.13.3}
\contentsline {subsection}{\numberline {45.13.4}\leavevmode {\color {Chapter }LGLayers}}{682}{subsection.45.13.4}
\contentsline {subsection}{\numberline {45.13.5}\leavevmode {\color {Chapter }LGFirst}}{682}{subsection.45.13.5}
\contentsline {subsection}{\numberline {45.13.6}\leavevmode {\color {Chapter }LGLength}}{682}{subsection.45.13.6}
\contentsline {subsection}{\numberline {45.13.7}\leavevmode {\color {Chapter }IsInducedPcgsWrtSpecialPcgs}}{683}{subsection.45.13.7}
\contentsline {subsection}{\numberline {45.13.8}\leavevmode {\color {Chapter }InducedPcgsWrtSpecialPcgs}}{683}{subsection.45.13.8}
\contentsline {section}{\numberline {45.14}\leavevmode {\color {Chapter }Action on Subfactors Defined by a Pcgs}}{683}{section.45.14}
\contentsline {subsection}{\numberline {45.14.1}\leavevmode {\color {Chapter }VectorSpaceByPcgsOfElementaryAbelianGroup}}{683}{subsection.45.14.1}
\contentsline {subsection}{\numberline {45.14.2}\leavevmode {\color {Chapter }LinearAction}}{684}{subsection.45.14.2}
\contentsline {subsection}{\numberline {45.14.3}\leavevmode {\color {Chapter }LinearActionLayer}}{684}{subsection.45.14.3}
\contentsline {subsection}{\numberline {45.14.4}\leavevmode {\color {Chapter }AffineAction}}{684}{subsection.45.14.4}
\contentsline {subsection}{\numberline {45.14.5}\leavevmode {\color {Chapter }AffineActionLayer}}{684}{subsection.45.14.5}
\contentsline {section}{\numberline {45.15}\leavevmode {\color {Chapter }Orbit Stabilizer Methods for Polycyclic Groups}}{685}{section.45.15}
\contentsline {subsection}{\numberline {45.15.1}\leavevmode {\color {Chapter }StabilizerPcgs}}{685}{subsection.45.15.1}
\contentsline {subsection}{\numberline {45.15.2}\leavevmode {\color {Chapter }Pcgs{\textunderscore }OrbitStabilizer}}{685}{subsection.45.15.2}
\contentsline {section}{\numberline {45.16}\leavevmode {\color {Chapter }Operations which have Special Methods for Groups with Pcgs}}{685}{section.45.16}
\contentsline {section}{\numberline {45.17}\leavevmode {\color {Chapter }Conjugacy Classes in Solvable Groups}}{685}{section.45.17}
\contentsline {subsection}{\numberline {45.17.1}\leavevmode {\color {Chapter }ClassesSolvableGroup}}{686}{subsection.45.17.1}
\contentsline {subsection}{\numberline {45.17.2}\leavevmode {\color {Chapter }CentralizerSizeLimitConsiderFunction}}{686}{subsection.45.17.2}
\contentsline {chapter}{\numberline {46}\leavevmode {\color {Chapter }Pc Groups}}{687}{chapter.46}
\contentsline {section}{\numberline {46.1}\leavevmode {\color {Chapter }The Family Pcgs}}{688}{section.46.1}
\contentsline {subsection}{\numberline {46.1.1}\leavevmode {\color {Chapter }FamilyPcgs}}{688}{subsection.46.1.1}
\contentsline {subsection}{\numberline {46.1.2}\leavevmode {\color {Chapter }IsFamilyPcgs}}{688}{subsection.46.1.2}
\contentsline {subsection}{\numberline {46.1.3}\leavevmode {\color {Chapter }InducedPcgsWrtFamilyPcgs}}{688}{subsection.46.1.3}
\contentsline {subsection}{\numberline {46.1.4}\leavevmode {\color {Chapter }IsParentPcgsFamilyPcgs}}{688}{subsection.46.1.4}
\contentsline {section}{\numberline {46.2}\leavevmode {\color {Chapter }Elements of Pc Groups}}{689}{section.46.2}
\contentsline {subsection}{\numberline {46.2.1}\leavevmode {\color {Chapter }Comparison of elements of pc groups}}{689}{subsection.46.2.1}
\contentsline {subsection}{\numberline {46.2.2}\leavevmode {\color {Chapter }Arithmetic operations for elements of pc groups}}{689}{subsection.46.2.2}
\contentsline {section}{\numberline {46.3}\leavevmode {\color {Chapter }Pc Groups versus Fp Groups}}{689}{section.46.3}
\contentsline {subsection}{\numberline {46.3.1}\leavevmode {\color {Chapter }IsPcGroup}}{689}{subsection.46.3.1}
\contentsline {subsection}{\numberline {46.3.2}\leavevmode {\color {Chapter }IsomorphismFpGroupByPcgs}}{690}{subsection.46.3.2}
\contentsline {section}{\numberline {46.4}\leavevmode {\color {Chapter }Constructing Pc Groups}}{690}{section.46.4}
\contentsline {subsection}{\numberline {46.4.1}\leavevmode {\color {Chapter }PcGroupFpGroup}}{690}{subsection.46.4.1}
\contentsline {subsection}{\numberline {46.4.2}\leavevmode {\color {Chapter }SingleCollector}}{691}{subsection.46.4.2}
\contentsline {subsection}{\numberline {46.4.3}\leavevmode {\color {Chapter }SetConjugate}}{691}{subsection.46.4.3}
\contentsline {subsection}{\numberline {46.4.4}\leavevmode {\color {Chapter }SetCommutator}}{691}{subsection.46.4.4}
\contentsline {subsection}{\numberline {46.4.5}\leavevmode {\color {Chapter }SetPower}}{691}{subsection.46.4.5}
\contentsline {subsection}{\numberline {46.4.6}\leavevmode {\color {Chapter }GroupByRws}}{692}{subsection.46.4.6}
\contentsline {subsection}{\numberline {46.4.7}\leavevmode {\color {Chapter }IsConfluent (for pc groups)}}{692}{subsection.46.4.7}
\contentsline {subsection}{\numberline {46.4.8}\leavevmode {\color {Chapter }IsomorphismRefinedPcGroup}}{692}{subsection.46.4.8}
\contentsline {subsection}{\numberline {46.4.9}\leavevmode {\color {Chapter }RefinedPcGroup}}{692}{subsection.46.4.9}
\contentsline {section}{\numberline {46.5}\leavevmode {\color {Chapter }Computing Pc Groups}}{693}{section.46.5}
\contentsline {subsection}{\numberline {46.5.1}\leavevmode {\color {Chapter }PcGroupWithPcgs}}{693}{subsection.46.5.1}
\contentsline {subsection}{\numberline {46.5.2}\leavevmode {\color {Chapter }IsomorphismPcGroup}}{693}{subsection.46.5.2}
\contentsline {subsection}{\numberline {46.5.3}\leavevmode {\color {Chapter }IsomorphismSpecialPcGroup}}{694}{subsection.46.5.3}
\contentsline {section}{\numberline {46.6}\leavevmode {\color {Chapter }Saving a Pc Group}}{694}{section.46.6}
\contentsline {subsection}{\numberline {46.6.1}\leavevmode {\color {Chapter }GapInputPcGroup}}{694}{subsection.46.6.1}
\contentsline {section}{\numberline {46.7}\leavevmode {\color {Chapter }Operations for Pc Groups}}{694}{section.46.7}
\contentsline {section}{\numberline {46.8}\leavevmode {\color {Chapter }$2$-Cohomology and Extensions}}{694}{section.46.8}
\contentsline {subsection}{\numberline {46.8.1}\leavevmode {\color {Chapter }TwoCoboundaries}}{695}{subsection.46.8.1}
\contentsline {subsection}{\numberline {46.8.2}\leavevmode {\color {Chapter }TwoCocycles}}{695}{subsection.46.8.2}
\contentsline {subsection}{\numberline {46.8.3}\leavevmode {\color {Chapter }TwoCohomology}}{695}{subsection.46.8.3}
\contentsline {subsection}{\numberline {46.8.4}\leavevmode {\color {Chapter }Extensions}}{696}{subsection.46.8.4}
\contentsline {subsection}{\numberline {46.8.5}\leavevmode {\color {Chapter }Extension}}{696}{subsection.46.8.5}
\contentsline {subsection}{\numberline {46.8.6}\leavevmode {\color {Chapter }SplitExtension}}{696}{subsection.46.8.6}
\contentsline {subsection}{\numberline {46.8.7}\leavevmode {\color {Chapter }ModuleOfExtension}}{696}{subsection.46.8.7}
\contentsline {subsection}{\numberline {46.8.8}\leavevmode {\color {Chapter }CompatiblePairs}}{697}{subsection.46.8.8}
\contentsline {subsection}{\numberline {46.8.9}\leavevmode {\color {Chapter }ExtensionRepresentatives}}{697}{subsection.46.8.9}
\contentsline {subsection}{\numberline {46.8.10}\leavevmode {\color {Chapter }SplitExtension (with specified homomorphism)}}{698}{subsection.46.8.10}
\contentsline {section}{\numberline {46.9}\leavevmode {\color {Chapter }Coding a Pc Presentation}}{698}{section.46.9}
\contentsline {subsection}{\numberline {46.9.1}\leavevmode {\color {Chapter }CodePcgs}}{698}{subsection.46.9.1}
\contentsline {subsection}{\numberline {46.9.2}\leavevmode {\color {Chapter }CodePcGroup}}{699}{subsection.46.9.2}
\contentsline {subsection}{\numberline {46.9.3}\leavevmode {\color {Chapter }PcGroupCode}}{699}{subsection.46.9.3}
\contentsline {section}{\numberline {46.10}\leavevmode {\color {Chapter }Random Isomorphism Testing}}{699}{section.46.10}
\contentsline {subsection}{\numberline {46.10.1}\leavevmode {\color {Chapter }RandomIsomorphismTest}}{699}{subsection.46.10.1}
\contentsline {chapter}{\numberline {47}\leavevmode {\color {Chapter }Finitely Presented Groups}}{700}{chapter.47}
\contentsline {section}{\numberline {47.1}\leavevmode {\color {Chapter }IsSubgroupFpGroup and IsFpGroup}}{701}{section.47.1}
\contentsline {subsection}{\numberline {47.1.1}\leavevmode {\color {Chapter }IsSubgroupFpGroup}}{701}{subsection.47.1.1}
\contentsline {subsection}{\numberline {47.1.2}\leavevmode {\color {Chapter }IsFpGroup}}{701}{subsection.47.1.2}
\contentsline {subsection}{\numberline {47.1.3}\leavevmode {\color {Chapter }InfoFpGroup}}{702}{subsection.47.1.3}
\contentsline {section}{\numberline {47.2}\leavevmode {\color {Chapter }Creating Finitely Presented Groups}}{702}{section.47.2}
\contentsline {subsection}{\numberline {47.2.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }/ (for a free group and a list of elements)}}{702}{subsection.47.2.1}
\contentsline {subsection}{\numberline {47.2.2}\leavevmode {\color {Chapter }FactorGroupFpGroupByRels}}{702}{subsection.47.2.2}
\contentsline {subsection}{\numberline {47.2.3}\leavevmode {\color {Chapter }ParseRelators}}{702}{subsection.47.2.3}
\contentsline {subsection}{\numberline {47.2.4}\leavevmode {\color {Chapter }StringFactorizationWord}}{703}{subsection.47.2.4}
\contentsline {section}{\numberline {47.3}\leavevmode {\color {Chapter }Comparison of Elements of Finitely Presented Groups}}{703}{section.47.3}
\contentsline {subsection}{\numberline {47.3.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= (for two elements in a f.p. group)}}{703}{subsection.47.3.1}
\contentsline {subsection}{\numberline {47.3.2}\leavevmode {\color {Chapter }\texttt {\char 92\relax }{\textless } (for two elements in a f.p. group)}}{704}{subsection.47.3.2}
\contentsline {subsection}{\numberline {47.3.3}\leavevmode {\color {Chapter }FpElmComparisonMethod}}{704}{subsection.47.3.3}
\contentsline {subsection}{\numberline {47.3.4}\leavevmode {\color {Chapter }SetReducedMultiplication}}{704}{subsection.47.3.4}
\contentsline {section}{\numberline {47.4}\leavevmode {\color {Chapter }Preimages in the Free Group}}{704}{section.47.4}
\contentsline {subsection}{\numberline {47.4.1}\leavevmode {\color {Chapter }FreeGroupOfFpGroup}}{704}{subsection.47.4.1}
\contentsline {subsection}{\numberline {47.4.2}\leavevmode {\color {Chapter }FreeGeneratorsOfFpGroup}}{704}{subsection.47.4.2}
\contentsline {subsection}{\numberline {47.4.3}\leavevmode {\color {Chapter }RelatorsOfFpGroup}}{705}{subsection.47.4.3}
\contentsline {subsection}{\numberline {47.4.4}\leavevmode {\color {Chapter }UnderlyingElement (fp group elements)}}{705}{subsection.47.4.4}
\contentsline {subsection}{\numberline {47.4.5}\leavevmode {\color {Chapter }ElementOfFpGroup}}{705}{subsection.47.4.5}
\contentsline {section}{\numberline {47.5}\leavevmode {\color {Chapter }Operations for Finitely Presented Groups}}{706}{section.47.5}
\contentsline {subsection}{\numberline {47.5.1}\leavevmode {\color {Chapter }PseudoRandom (for finitely presented groups)}}{706}{subsection.47.5.1}
\contentsline {section}{\numberline {47.6}\leavevmode {\color {Chapter }Coset Tables and Coset Enumeration}}{706}{section.47.6}
\contentsline {subsection}{\numberline {47.6.1}\leavevmode {\color {Chapter }CosetTable}}{707}{subsection.47.6.1}
\contentsline {subsection}{\numberline {47.6.2}\leavevmode {\color {Chapter }TracedCosetFpGroup}}{707}{subsection.47.6.2}
\contentsline {subsection}{\numberline {47.6.3}\leavevmode {\color {Chapter }FactorCosetAction (for fp groups)}}{708}{subsection.47.6.3}
\contentsline {subsection}{\numberline {47.6.4}\leavevmode {\color {Chapter }CosetTableBySubgroup}}{708}{subsection.47.6.4}
\contentsline {subsection}{\numberline {47.6.5}\leavevmode {\color {Chapter }CosetTableFromGensAndRels}}{708}{subsection.47.6.5}
\contentsline {subsection}{\numberline {47.6.6}\leavevmode {\color {Chapter }CosetTableDefaultMaxLimit}}{709}{subsection.47.6.6}
\contentsline {subsection}{\numberline {47.6.7}\leavevmode {\color {Chapter }CosetTableDefaultLimit}}{710}{subsection.47.6.7}
\contentsline {subsection}{\numberline {47.6.8}\leavevmode {\color {Chapter }MostFrequentGeneratorFpGroup}}{710}{subsection.47.6.8}
\contentsline {subsection}{\numberline {47.6.9}\leavevmode {\color {Chapter }IndicesInvolutaryGenerators}}{710}{subsection.47.6.9}
\contentsline {section}{\numberline {47.7}\leavevmode {\color {Chapter }Standardization of coset tables}}{710}{section.47.7}
\contentsline {subsection}{\numberline {47.7.1}\leavevmode {\color {Chapter }CosetTableStandard}}{711}{subsection.47.7.1}
\contentsline {subsection}{\numberline {47.7.2}\leavevmode {\color {Chapter }StandardizeTable}}{711}{subsection.47.7.2}
\contentsline {section}{\numberline {47.8}\leavevmode {\color {Chapter }Coset tables for subgroups in the whole group}}{712}{section.47.8}
\contentsline {subsection}{\numberline {47.8.1}\leavevmode {\color {Chapter }CosetTableInWholeGroup}}{712}{subsection.47.8.1}
\contentsline {subsection}{\numberline {47.8.2}\leavevmode {\color {Chapter }SubgroupOfWholeGroupByCosetTable}}{712}{subsection.47.8.2}
\contentsline {section}{\numberline {47.9}\leavevmode {\color {Chapter }Augmented Coset Tables and Rewriting}}{712}{section.47.9}
\contentsline {subsection}{\numberline {47.9.1}\leavevmode {\color {Chapter }AugmentedCosetTableInWholeGroup}}{712}{subsection.47.9.1}
\contentsline {subsection}{\numberline {47.9.2}\leavevmode {\color {Chapter }AugmentedCosetTableMtc}}{712}{subsection.47.9.2}
\contentsline {subsection}{\numberline {47.9.3}\leavevmode {\color {Chapter }AugmentedCosetTableRrs}}{713}{subsection.47.9.3}
\contentsline {subsection}{\numberline {47.9.4}\leavevmode {\color {Chapter }RewriteWord}}{713}{subsection.47.9.4}
\contentsline {section}{\numberline {47.10}\leavevmode {\color {Chapter }Low Index Subgroups}}{713}{section.47.10}
\contentsline {subsection}{\numberline {47.10.1}\leavevmode {\color {Chapter }LowIndexSubgroupsFpGroupIterator}}{713}{subsection.47.10.1}
\contentsline {section}{\numberline {47.11}\leavevmode {\color {Chapter }Converting Groups to Finitely Presented Groups}}{714}{section.47.11}
\contentsline {subsection}{\numberline {47.11.1}\leavevmode {\color {Chapter }IsomorphismFpGroup}}{714}{subsection.47.11.1}
\contentsline {subsection}{\numberline {47.11.2}\leavevmode {\color {Chapter }IsomorphismFpGroupByGenerators}}{715}{subsection.47.11.2}
\contentsline {section}{\numberline {47.12}\leavevmode {\color {Chapter }New Presentations and Presentations for Subgroups}}{717}{section.47.12}
\contentsline {subsection}{\numberline {47.12.1}\leavevmode {\color {Chapter }IsomorphismSimplifiedFpGroup}}{718}{subsection.47.12.1}
\contentsline {section}{\numberline {47.13}\leavevmode {\color {Chapter }Preimages under Homomorphisms from an FpGroup}}{718}{section.47.13}
\contentsline {subsection}{\numberline {47.13.1}\leavevmode {\color {Chapter }SubgroupOfWholeGroupByQuotientSubgroup}}{719}{subsection.47.13.1}
\contentsline {subsection}{\numberline {47.13.2}\leavevmode {\color {Chapter }IsSubgroupOfWholeGroupByQuotientRep}}{719}{subsection.47.13.2}
\contentsline {subsection}{\numberline {47.13.3}\leavevmode {\color {Chapter }AsSubgroupOfWholeGroupByQuotient}}{719}{subsection.47.13.3}
\contentsline {subsection}{\numberline {47.13.4}\leavevmode {\color {Chapter }DefiningQuotientHomomorphism}}{719}{subsection.47.13.4}
\contentsline {section}{\numberline {47.14}\leavevmode {\color {Chapter }Quotient Methods}}{719}{section.47.14}
\contentsline {subsection}{\numberline {47.14.1}\leavevmode {\color {Chapter }PQuotient}}{720}{subsection.47.14.1}
\contentsline {subsection}{\numberline {47.14.2}\leavevmode {\color {Chapter }EpimorphismQuotientSystem}}{720}{subsection.47.14.2}
\contentsline {subsection}{\numberline {47.14.3}\leavevmode {\color {Chapter }EpimorphismPGroup}}{721}{subsection.47.14.3}
\contentsline {subsection}{\numberline {47.14.4}\leavevmode {\color {Chapter }EpimorphismNilpotentQuotient}}{721}{subsection.47.14.4}
\contentsline {subsection}{\numberline {47.14.5}\leavevmode {\color {Chapter }SolvableQuotient (for a f.p. group and a size)}}{721}{subsection.47.14.5}
\contentsline {subsection}{\numberline {47.14.6}\leavevmode {\color {Chapter }EpimorphismSolvableQuotient}}{722}{subsection.47.14.6}
\contentsline {subsection}{\numberline {47.14.7}\leavevmode {\color {Chapter }LargerQuotientBySubgroupAbelianization}}{722}{subsection.47.14.7}
\contentsline {section}{\numberline {47.15}\leavevmode {\color {Chapter }Abelian Invariants for Subgroups}}{722}{section.47.15}
\contentsline {subsection}{\numberline {47.15.1}\leavevmode {\color {Chapter }AbelianInvariantsSubgroupFpGroup}}{723}{subsection.47.15.1}
\contentsline {subsection}{\numberline {47.15.2}\leavevmode {\color {Chapter }AbelianInvariantsSubgroupFpGroupMtc}}{723}{subsection.47.15.2}
\contentsline {subsection}{\numberline {47.15.3}\leavevmode {\color {Chapter }AbelianInvariantsSubgroupFpGroupRrs}}{723}{subsection.47.15.3}
\contentsline {subsection}{\numberline {47.15.4}\leavevmode {\color {Chapter }AbelianInvariantsNormalClosureFpGroup}}{723}{subsection.47.15.4}
\contentsline {subsection}{\numberline {47.15.5}\leavevmode {\color {Chapter }AbelianInvariantsNormalClosureFpGroupRrs}}{723}{subsection.47.15.5}
\contentsline {section}{\numberline {47.16}\leavevmode {\color {Chapter }Testing Finiteness of Finitely Presented Groups}}{724}{section.47.16}
\contentsline {subsection}{\numberline {47.16.1}\leavevmode {\color {Chapter }IsInfiniteAbelianizationGroup}}{725}{subsection.47.16.1}
\contentsline {subsection}{\numberline {47.16.2}\leavevmode {\color {Chapter }NewmanInfinityCriterion}}{725}{subsection.47.16.2}
\contentsline {chapter}{\numberline {48}\leavevmode {\color {Chapter }Presentations and Tietze Transformations}}{726}{chapter.48}
\contentsline {section}{\numberline {48.1}\leavevmode {\color {Chapter }Creating Presentations}}{726}{section.48.1}
\contentsline {subsection}{\numberline {48.1.1}\leavevmode {\color {Chapter }PresentationFpGroup}}{726}{subsection.48.1.1}
\contentsline {subsection}{\numberline {48.1.2}\leavevmode {\color {Chapter }TzSort}}{727}{subsection.48.1.2}
\contentsline {subsection}{\numberline {48.1.3}\leavevmode {\color {Chapter }GeneratorsOfPresentation}}{727}{subsection.48.1.3}
\contentsline {subsection}{\numberline {48.1.4}\leavevmode {\color {Chapter }FpGroupPresentation}}{727}{subsection.48.1.4}
\contentsline {subsection}{\numberline {48.1.5}\leavevmode {\color {Chapter }PresentationViaCosetTable}}{727}{subsection.48.1.5}
\contentsline {subsection}{\numberline {48.1.6}\leavevmode {\color {Chapter }SimplifiedFpGroup}}{729}{subsection.48.1.6}
\contentsline {section}{\numberline {48.2}\leavevmode {\color {Chapter }Subgroup Presentations}}{729}{section.48.2}
\contentsline {subsection}{\numberline {48.2.1}\leavevmode {\color {Chapter }PresentationSubgroup}}{729}{subsection.48.2.1}
\contentsline {subsection}{\numberline {48.2.2}\leavevmode {\color {Chapter }PresentationSubgroupRrs}}{729}{subsection.48.2.2}
\contentsline {subsection}{\numberline {48.2.3}\leavevmode {\color {Chapter }PrimaryGeneratorWords}}{731}{subsection.48.2.3}
\contentsline {subsection}{\numberline {48.2.4}\leavevmode {\color {Chapter }PresentationSubgroupMtc}}{731}{subsection.48.2.4}
\contentsline {subsection}{\numberline {48.2.5}\leavevmode {\color {Chapter }PresentationNormalClosureRrs}}{733}{subsection.48.2.5}
\contentsline {subsection}{\numberline {48.2.6}\leavevmode {\color {Chapter }PresentationNormalClosure}}{733}{subsection.48.2.6}
\contentsline {section}{\numberline {48.3}\leavevmode {\color {Chapter }Relators in a Presentation}}{733}{section.48.3}
\contentsline {subsection}{\numberline {48.3.1}\leavevmode {\color {Chapter }TietzeWordAbstractWord}}{733}{subsection.48.3.1}
\contentsline {subsection}{\numberline {48.3.2}\leavevmode {\color {Chapter }AbstractWordTietzeWord}}{733}{subsection.48.3.2}
\contentsline {section}{\numberline {48.4}\leavevmode {\color {Chapter }Printing Presentations}}{734}{section.48.4}
\contentsline {subsection}{\numberline {48.4.1}\leavevmode {\color {Chapter }TzPrintGenerators}}{734}{subsection.48.4.1}
\contentsline {subsection}{\numberline {48.4.2}\leavevmode {\color {Chapter }TzPrintRelators}}{734}{subsection.48.4.2}
\contentsline {subsection}{\numberline {48.4.3}\leavevmode {\color {Chapter }TzPrintLengths}}{734}{subsection.48.4.3}
\contentsline {subsection}{\numberline {48.4.4}\leavevmode {\color {Chapter }TzPrintStatus}}{735}{subsection.48.4.4}
\contentsline {subsection}{\numberline {48.4.5}\leavevmode {\color {Chapter }TzPrintPresentation}}{735}{subsection.48.4.5}
\contentsline {subsection}{\numberline {48.4.6}\leavevmode {\color {Chapter }TzPrint}}{735}{subsection.48.4.6}
\contentsline {subsection}{\numberline {48.4.7}\leavevmode {\color {Chapter }TzPrintPairs}}{735}{subsection.48.4.7}
\contentsline {section}{\numberline {48.5}\leavevmode {\color {Chapter }Changing Presentations}}{736}{section.48.5}
\contentsline {subsection}{\numberline {48.5.1}\leavevmode {\color {Chapter }AddGenerator}}{736}{subsection.48.5.1}
\contentsline {subsection}{\numberline {48.5.2}\leavevmode {\color {Chapter }TzNewGenerator}}{736}{subsection.48.5.2}
\contentsline {subsection}{\numberline {48.5.3}\leavevmode {\color {Chapter }AddRelator}}{737}{subsection.48.5.3}
\contentsline {subsection}{\numberline {48.5.4}\leavevmode {\color {Chapter }RemoveRelator}}{737}{subsection.48.5.4}
\contentsline {section}{\numberline {48.6}\leavevmode {\color {Chapter }Tietze Transformations}}{737}{section.48.6}
\contentsline {subsection}{\numberline {48.6.1}\leavevmode {\color {Chapter }TzGo}}{737}{subsection.48.6.1}
\contentsline {subsection}{\numberline {48.6.2}\leavevmode {\color {Chapter }SimplifyPresentation}}{737}{subsection.48.6.2}
\contentsline {subsection}{\numberline {48.6.3}\leavevmode {\color {Chapter }TzGoGo}}{738}{subsection.48.6.3}
\contentsline {section}{\numberline {48.7}\leavevmode {\color {Chapter }Elementary Tietze Transformations}}{740}{section.48.7}
\contentsline {subsection}{\numberline {48.7.1}\leavevmode {\color {Chapter }TzEliminate}}{740}{subsection.48.7.1}
\contentsline {subsection}{\numberline {48.7.2}\leavevmode {\color {Chapter }TzSearch}}{740}{subsection.48.7.2}
\contentsline {subsection}{\numberline {48.7.3}\leavevmode {\color {Chapter }TzSearchEqual}}{741}{subsection.48.7.3}
\contentsline {subsection}{\numberline {48.7.4}\leavevmode {\color {Chapter }TzFindCyclicJoins}}{741}{subsection.48.7.4}
\contentsline {section}{\numberline {48.8}\leavevmode {\color {Chapter }Tietze Transformations that introduce new Generators}}{742}{section.48.8}
\contentsline {subsection}{\numberline {48.8.1}\leavevmode {\color {Chapter }TzSubstitute}}{742}{subsection.48.8.1}
\contentsline {subsection}{\numberline {48.8.2}\leavevmode {\color {Chapter }TzSubstituteCyclicJoins}}{745}{subsection.48.8.2}
\contentsline {section}{\numberline {48.9}\leavevmode {\color {Chapter }Tracing generator images through Tietze transformations}}{745}{section.48.9}
\contentsline {subsection}{\numberline {48.9.1}\leavevmode {\color {Chapter }TzInitGeneratorImages}}{746}{subsection.48.9.1}
\contentsline {subsection}{\numberline {48.9.2}\leavevmode {\color {Chapter }OldGeneratorsOfPresentation}}{746}{subsection.48.9.2}
\contentsline {subsection}{\numberline {48.9.3}\leavevmode {\color {Chapter }TzImagesOldGens}}{746}{subsection.48.9.3}
\contentsline {subsection}{\numberline {48.9.4}\leavevmode {\color {Chapter }TzPreImagesNewGens}}{746}{subsection.48.9.4}
\contentsline {subsection}{\numberline {48.9.5}\leavevmode {\color {Chapter }TzPrintGeneratorImages}}{747}{subsection.48.9.5}
\contentsline {section}{\numberline {48.10}\leavevmode {\color {Chapter }The Decoding Tree Procedure}}{747}{section.48.10}
\contentsline {subsection}{\numberline {48.10.1}\leavevmode {\color {Chapter }DecodeTree}}{747}{subsection.48.10.1}
\contentsline {section}{\numberline {48.11}\leavevmode {\color {Chapter }Tietze Options}}{750}{section.48.11}
\contentsline {subsection}{\numberline {48.11.1}\leavevmode {\color {Chapter }TzOptions}}{750}{subsection.48.11.1}
\contentsline {subsection}{\numberline {48.11.2}\leavevmode {\color {Chapter }TzPrintOptions}}{752}{subsection.48.11.2}
\contentsline {chapter}{\numberline {49}\leavevmode {\color {Chapter }Group Products}}{753}{chapter.49}
\contentsline {section}{\numberline {49.1}\leavevmode {\color {Chapter }Direct Products}}{753}{section.49.1}
\contentsline {subsection}{\numberline {49.1.1}\leavevmode {\color {Chapter }DirectProduct}}{753}{subsection.49.1.1}
\contentsline {section}{\numberline {49.2}\leavevmode {\color {Chapter }Semidirect Products}}{754}{section.49.2}
\contentsline {subsection}{\numberline {49.2.1}\leavevmode {\color {Chapter }SemidirectProduct}}{754}{subsection.49.2.1}
\contentsline {section}{\numberline {49.3}\leavevmode {\color {Chapter }Subdirect Products}}{756}{section.49.3}
\contentsline {subsection}{\numberline {49.3.1}\leavevmode {\color {Chapter }SubdirectProduct}}{756}{subsection.49.3.1}
\contentsline {subsection}{\numberline {49.3.2}\leavevmode {\color {Chapter }SubdirectProducts}}{756}{subsection.49.3.2}
\contentsline {section}{\numberline {49.4}\leavevmode {\color {Chapter }Wreath Products}}{756}{section.49.4}
\contentsline {subsection}{\numberline {49.4.1}\leavevmode {\color {Chapter }WreathProduct}}{757}{subsection.49.4.1}
\contentsline {subsection}{\numberline {49.4.2}\leavevmode {\color {Chapter }WreathProductImprimitiveAction}}{757}{subsection.49.4.2}
\contentsline {subsection}{\numberline {49.4.3}\leavevmode {\color {Chapter }WreathProductProductAction}}{758}{subsection.49.4.3}
\contentsline {subsection}{\numberline {49.4.4}\leavevmode {\color {Chapter }KuKGenerators}}{758}{subsection.49.4.4}
\contentsline {section}{\numberline {49.5}\leavevmode {\color {Chapter }Free Products}}{758}{section.49.5}
\contentsline {subsection}{\numberline {49.5.1}\leavevmode {\color {Chapter }FreeProduct}}{759}{subsection.49.5.1}
\contentsline {section}{\numberline {49.6}\leavevmode {\color {Chapter }Embeddings and Projections for Group Products}}{759}{section.49.6}
\contentsline {subsection}{\numberline {49.6.1}\leavevmode {\color {Chapter }Embedding (for group products)}}{759}{subsection.49.6.1}
\contentsline {subsection}{\numberline {49.6.2}\leavevmode {\color {Chapter }Projection (for group products)}}{759}{subsection.49.6.2}
\contentsline {chapter}{\numberline {50}\leavevmode {\color {Chapter }Group Libraries}}{760}{chapter.50}
\contentsline {section}{\numberline {50.1}\leavevmode {\color {Chapter }Basic Groups}}{761}{section.50.1}
\contentsline {subsection}{\numberline {50.1.1}\leavevmode {\color {Chapter }TrivialGroup}}{761}{subsection.50.1.1}
\contentsline {subsection}{\numberline {50.1.2}\leavevmode {\color {Chapter }CyclicGroup}}{761}{subsection.50.1.2}
\contentsline {subsection}{\numberline {50.1.3}\leavevmode {\color {Chapter }AbelianGroup}}{762}{subsection.50.1.3}
\contentsline {subsection}{\numberline {50.1.4}\leavevmode {\color {Chapter }ElementaryAbelianGroup}}{762}{subsection.50.1.4}
\contentsline {subsection}{\numberline {50.1.5}\leavevmode {\color {Chapter }FreeAbelianGroup}}{762}{subsection.50.1.5}
\contentsline {subsection}{\numberline {50.1.6}\leavevmode {\color {Chapter }DihedralGroup}}{762}{subsection.50.1.6}
\contentsline {subsection}{\numberline {50.1.7}\leavevmode {\color {Chapter }IsDihedralGroup}}{763}{subsection.50.1.7}
\contentsline {subsection}{\numberline {50.1.8}\leavevmode {\color {Chapter }DicyclicGroup}}{763}{subsection.50.1.8}
\contentsline {subsection}{\numberline {50.1.9}\leavevmode {\color {Chapter }IsGeneralisedQuaternionGroup}}{763}{subsection.50.1.9}
\contentsline {subsection}{\numberline {50.1.10}\leavevmode {\color {Chapter }ExtraspecialGroup}}{764}{subsection.50.1.10}
\contentsline {subsection}{\numberline {50.1.11}\leavevmode {\color {Chapter }AlternatingGroup}}{764}{subsection.50.1.11}
\contentsline {subsection}{\numberline {50.1.12}\leavevmode {\color {Chapter }SymmetricGroup}}{764}{subsection.50.1.12}
\contentsline {subsection}{\numberline {50.1.13}\leavevmode {\color {Chapter }MathieuGroup}}{765}{subsection.50.1.13}
\contentsline {subsection}{\numberline {50.1.14}\leavevmode {\color {Chapter }SuzukiGroup}}{765}{subsection.50.1.14}
\contentsline {subsection}{\numberline {50.1.15}\leavevmode {\color {Chapter }ReeGroup}}{765}{subsection.50.1.15}
\contentsline {section}{\numberline {50.2}\leavevmode {\color {Chapter }Classical Groups}}{765}{section.50.2}
\contentsline {subsection}{\numberline {50.2.1}\leavevmode {\color {Chapter }GeneralLinearGroup}}{766}{subsection.50.2.1}
\contentsline {subsection}{\numberline {50.2.2}\leavevmode {\color {Chapter }SpecialLinearGroup}}{767}{subsection.50.2.2}
\contentsline {subsection}{\numberline {50.2.3}\leavevmode {\color {Chapter }GeneralUnitaryGroup}}{767}{subsection.50.2.3}
\contentsline {subsection}{\numberline {50.2.4}\leavevmode {\color {Chapter }SpecialUnitaryGroup}}{768}{subsection.50.2.4}
\contentsline {subsection}{\numberline {50.2.5}\leavevmode {\color {Chapter }SymplecticGroup}}{768}{subsection.50.2.5}
\contentsline {subsection}{\numberline {50.2.6}\leavevmode {\color {Chapter }GeneralOrthogonalGroup}}{768}{subsection.50.2.6}
\contentsline {subsection}{\numberline {50.2.7}\leavevmode {\color {Chapter }SpecialOrthogonalGroup}}{769}{subsection.50.2.7}
\contentsline {subsection}{\numberline {50.2.8}\leavevmode {\color {Chapter }Omega (construct an orthogonal group)}}{769}{subsection.50.2.8}
\contentsline {subsection}{\numberline {50.2.9}\leavevmode {\color {Chapter }GeneralSemilinearGroup}}{770}{subsection.50.2.9}
\contentsline {subsection}{\numberline {50.2.10}\leavevmode {\color {Chapter }SpecialSemilinearGroup}}{770}{subsection.50.2.10}
\contentsline {subsection}{\numberline {50.2.11}\leavevmode {\color {Chapter }ProjectiveGeneralLinearGroup}}{770}{subsection.50.2.11}
\contentsline {subsection}{\numberline {50.2.12}\leavevmode {\color {Chapter }ProjectiveSpecialLinearGroup}}{770}{subsection.50.2.12}
\contentsline {subsection}{\numberline {50.2.13}\leavevmode {\color {Chapter }ProjectiveGeneralUnitaryGroup}}{771}{subsection.50.2.13}
\contentsline {subsection}{\numberline {50.2.14}\leavevmode {\color {Chapter }ProjectiveSpecialUnitaryGroup}}{771}{subsection.50.2.14}
\contentsline {subsection}{\numberline {50.2.15}\leavevmode {\color {Chapter }ProjectiveSymplecticGroup}}{771}{subsection.50.2.15}
\contentsline {subsection}{\numberline {50.2.16}\leavevmode {\color {Chapter }ProjectiveOmega}}{771}{subsection.50.2.16}
\contentsline {section}{\numberline {50.3}\leavevmode {\color {Chapter }Conjugacy Classes in Classical Groups}}{772}{section.50.3}
\contentsline {subsection}{\numberline {50.3.1}\leavevmode {\color {Chapter }NrConjugacyClassesGL}}{772}{subsection.50.3.1}
\contentsline {section}{\numberline {50.4}\leavevmode {\color {Chapter }Constructors for Basic Groups}}{773}{section.50.4}
\contentsline {section}{\numberline {50.5}\leavevmode {\color {Chapter }Selection Functions}}{773}{section.50.5}
\contentsline {section}{\numberline {50.6}\leavevmode {\color {Chapter }Finite Perfect Groups}}{774}{section.50.6}
\contentsline {subsection}{\numberline {50.6.1}\leavevmode {\color {Chapter }SizesPerfectGroups}}{775}{subsection.50.6.1}
\contentsline {subsection}{\numberline {50.6.2}\leavevmode {\color {Chapter }PerfectGroup}}{775}{subsection.50.6.2}
\contentsline {subsection}{\numberline {50.6.3}\leavevmode {\color {Chapter }PerfectIdentification}}{775}{subsection.50.6.3}
\contentsline {subsection}{\numberline {50.6.4}\leavevmode {\color {Chapter }NumberPerfectGroups}}{776}{subsection.50.6.4}
\contentsline {subsection}{\numberline {50.6.5}\leavevmode {\color {Chapter }NumberPerfectLibraryGroups}}{776}{subsection.50.6.5}
\contentsline {subsection}{\numberline {50.6.6}\leavevmode {\color {Chapter }SizeNumbersPerfectGroups}}{776}{subsection.50.6.6}
\contentsline {subsection}{\numberline {50.6.7}\leavevmode {\color {Chapter }DisplayInformationPerfectGroups}}{777}{subsection.50.6.7}
\contentsline {subsection}{\numberline {50.6.8}\leavevmode {\color {Chapter }More about the Perfect Groups Library}}{778}{subsection.50.6.8}
\contentsline {section}{\numberline {50.7}\leavevmode {\color {Chapter }Irreducible Maximal Finite Integral Matrix Groups}}{780}{section.50.7}
\contentsline {subsection}{\numberline {50.7.1}\leavevmode {\color {Chapter }ImfNumberQQClasses}}{781}{subsection.50.7.1}
\contentsline {subsection}{\numberline {50.7.2}\leavevmode {\color {Chapter }DisplayImfInvariants}}{782}{subsection.50.7.2}
\contentsline {subsection}{\numberline {50.7.3}\leavevmode {\color {Chapter }ImfInvariants}}{784}{subsection.50.7.3}
\contentsline {subsection}{\numberline {50.7.4}\leavevmode {\color {Chapter }ImfMatrixGroup}}{785}{subsection.50.7.4}
\contentsline {subsection}{\numberline {50.7.5}\leavevmode {\color {Chapter }IsomorphismPermGroup (for Imf matrix groups)}}{787}{subsection.50.7.5}
\contentsline {subsection}{\numberline {50.7.6}\leavevmode {\color {Chapter }IsomorphismPermGroupImfGroup}}{788}{subsection.50.7.6}
\contentsline {chapter}{\numberline {51}\leavevmode {\color {Chapter }Semigroups and Monoids}}{789}{chapter.51}
\contentsline {section}{\numberline {51.1}\leavevmode {\color {Chapter }Semigroups}}{789}{section.51.1}
\contentsline {subsection}{\numberline {51.1.1}\leavevmode {\color {Chapter }IsSemigroup}}{789}{subsection.51.1.1}
\contentsline {subsection}{\numberline {51.1.2}\leavevmode {\color {Chapter }Semigroup}}{789}{subsection.51.1.2}
\contentsline {subsection}{\numberline {51.1.3}\leavevmode {\color {Chapter }Subsemigroup}}{789}{subsection.51.1.3}
\contentsline {subsection}{\numberline {51.1.4}\leavevmode {\color {Chapter }IsSubsemigroup}}{790}{subsection.51.1.4}
\contentsline {subsection}{\numberline {51.1.5}\leavevmode {\color {Chapter }SemigroupByGenerators}}{790}{subsection.51.1.5}
\contentsline {subsection}{\numberline {51.1.6}\leavevmode {\color {Chapter }AsSemigroup}}{790}{subsection.51.1.6}
\contentsline {subsection}{\numberline {51.1.7}\leavevmode {\color {Chapter }AsSubsemigroup}}{790}{subsection.51.1.7}
\contentsline {subsection}{\numberline {51.1.8}\leavevmode {\color {Chapter }GeneratorsOfSemigroup}}{791}{subsection.51.1.8}
\contentsline {subsection}{\numberline {51.1.9}\leavevmode {\color {Chapter }IsGeneratorsOfSemigroup}}{791}{subsection.51.1.9}
\contentsline {subsection}{\numberline {51.1.10}\leavevmode {\color {Chapter }FreeSemigroup}}{791}{subsection.51.1.10}
\contentsline {subsection}{\numberline {51.1.11}\leavevmode {\color {Chapter }SemigroupByMultiplicationTable}}{792}{subsection.51.1.11}
\contentsline {section}{\numberline {51.2}\leavevmode {\color {Chapter }Monoids}}{793}{section.51.2}
\contentsline {subsection}{\numberline {51.2.1}\leavevmode {\color {Chapter }IsMonoid}}{793}{subsection.51.2.1}
\contentsline {subsection}{\numberline {51.2.2}\leavevmode {\color {Chapter }Monoid}}{793}{subsection.51.2.2}
\contentsline {subsection}{\numberline {51.2.3}\leavevmode {\color {Chapter }Submonoid}}{793}{subsection.51.2.3}
\contentsline {subsection}{\numberline {51.2.4}\leavevmode {\color {Chapter }MonoidByGenerators}}{793}{subsection.51.2.4}
\contentsline {subsection}{\numberline {51.2.5}\leavevmode {\color {Chapter }AsMonoid}}{793}{subsection.51.2.5}
\contentsline {subsection}{\numberline {51.2.6}\leavevmode {\color {Chapter }AsSubmonoid}}{794}{subsection.51.2.6}
\contentsline {subsection}{\numberline {51.2.7}\leavevmode {\color {Chapter }GeneratorsOfMonoid}}{794}{subsection.51.2.7}
\contentsline {subsection}{\numberline {51.2.8}\leavevmode {\color {Chapter }TrivialSubmonoid}}{794}{subsection.51.2.8}
\contentsline {subsection}{\numberline {51.2.9}\leavevmode {\color {Chapter }FreeMonoid}}{794}{subsection.51.2.9}
\contentsline {subsection}{\numberline {51.2.10}\leavevmode {\color {Chapter }MonoidByMultiplicationTable}}{794}{subsection.51.2.10}
\contentsline {section}{\numberline {51.3}\leavevmode {\color {Chapter }Inverse semigroups and monoids}}{795}{section.51.3}
\contentsline {subsection}{\numberline {51.3.1}\leavevmode {\color {Chapter }InverseSemigroup}}{795}{subsection.51.3.1}
\contentsline {subsection}{\numberline {51.3.2}\leavevmode {\color {Chapter }InverseMonoid}}{795}{subsection.51.3.2}
\contentsline {subsection}{\numberline {51.3.3}\leavevmode {\color {Chapter }GeneratorsOfInverseSemigroup}}{796}{subsection.51.3.3}
\contentsline {subsection}{\numberline {51.3.4}\leavevmode {\color {Chapter }GeneratorsOfInverseMonoid}}{796}{subsection.51.3.4}
\contentsline {subsection}{\numberline {51.3.5}\leavevmode {\color {Chapter }IsInverseSubsemigroup}}{797}{subsection.51.3.5}
\contentsline {section}{\numberline {51.4}\leavevmode {\color {Chapter }Properties of Semigroups}}{797}{section.51.4}
\contentsline {subsection}{\numberline {51.4.1}\leavevmode {\color {Chapter }IsRegularSemigroup}}{797}{subsection.51.4.1}
\contentsline {subsection}{\numberline {51.4.2}\leavevmode {\color {Chapter }IsRegularSemigroupElement}}{797}{subsection.51.4.2}
\contentsline {subsection}{\numberline {51.4.3}\leavevmode {\color {Chapter }InversesOfSemigroupElement}}{797}{subsection.51.4.3}
\contentsline {subsection}{\numberline {51.4.4}\leavevmode {\color {Chapter }IsSimpleSemigroup}}{798}{subsection.51.4.4}
\contentsline {subsection}{\numberline {51.4.5}\leavevmode {\color {Chapter }IsZeroSimpleSemigroup}}{798}{subsection.51.4.5}
\contentsline {subsection}{\numberline {51.4.6}\leavevmode {\color {Chapter }IsZeroGroup}}{798}{subsection.51.4.6}
\contentsline {subsection}{\numberline {51.4.7}\leavevmode {\color {Chapter }IsReesCongruenceSemigroup}}{798}{subsection.51.4.7}
\contentsline {subsection}{\numberline {51.4.8}\leavevmode {\color {Chapter }IsInverseSemigroup}}{799}{subsection.51.4.8}
\contentsline {section}{\numberline {51.5}\leavevmode {\color {Chapter }Ideals of semigroups}}{799}{section.51.5}
\contentsline {subsection}{\numberline {51.5.1}\leavevmode {\color {Chapter }SemigroupIdealByGenerators}}{799}{subsection.51.5.1}
\contentsline {subsection}{\numberline {51.5.2}\leavevmode {\color {Chapter }ReesCongruenceOfSemigroupIdeal}}{799}{subsection.51.5.2}
\contentsline {subsection}{\numberline {51.5.3}\leavevmode {\color {Chapter }IsLeftSemigroupIdeal}}{799}{subsection.51.5.3}
\contentsline {section}{\numberline {51.6}\leavevmode {\color {Chapter }Congruences on semigroups}}{799}{section.51.6}
\contentsline {subsection}{\numberline {51.6.1}\leavevmode {\color {Chapter }IsSemigroupCongruence}}{800}{subsection.51.6.1}
\contentsline {subsection}{\numberline {51.6.2}\leavevmode {\color {Chapter }IsReesCongruence}}{800}{subsection.51.6.2}
\contentsline {section}{\numberline {51.7}\leavevmode {\color {Chapter }Quotients}}{800}{section.51.7}
\contentsline {subsection}{\numberline {51.7.1}\leavevmode {\color {Chapter }IsQuotientSemigroup}}{800}{subsection.51.7.1}
\contentsline {subsection}{\numberline {51.7.2}\leavevmode {\color {Chapter }HomomorphismQuotientSemigroup}}{800}{subsection.51.7.2}
\contentsline {subsection}{\numberline {51.7.3}\leavevmode {\color {Chapter }QuotientSemigroupPreimage}}{800}{subsection.51.7.3}
\contentsline {section}{\numberline {51.8}\leavevmode {\color {Chapter }Green's Relations}}{801}{section.51.8}
\contentsline {subsection}{\numberline {51.8.1}\leavevmode {\color {Chapter }GreensRRelation}}{801}{subsection.51.8.1}
\contentsline {subsection}{\numberline {51.8.2}\leavevmode {\color {Chapter }IsGreensRelation}}{801}{subsection.51.8.2}
\contentsline {subsection}{\numberline {51.8.3}\leavevmode {\color {Chapter }IsGreensClass}}{801}{subsection.51.8.3}
\contentsline {subsection}{\numberline {51.8.4}\leavevmode {\color {Chapter }IsGreensLessThanOrEqual}}{802}{subsection.51.8.4}
\contentsline {subsection}{\numberline {51.8.5}\leavevmode {\color {Chapter }RClassOfHClass}}{802}{subsection.51.8.5}
\contentsline {subsection}{\numberline {51.8.6}\leavevmode {\color {Chapter }EggBoxOfDClass}}{802}{subsection.51.8.6}
\contentsline {subsection}{\numberline {51.8.7}\leavevmode {\color {Chapter }DisplayEggBoxOfDClass}}{802}{subsection.51.8.7}
\contentsline {subsection}{\numberline {51.8.8}\leavevmode {\color {Chapter }GreensRClassOfElement}}{802}{subsection.51.8.8}
\contentsline {subsection}{\numberline {51.8.9}\leavevmode {\color {Chapter }GreensRClasses}}{802}{subsection.51.8.9}
\contentsline {subsection}{\numberline {51.8.10}\leavevmode {\color {Chapter }GroupHClassOfGreensDClass}}{803}{subsection.51.8.10}
\contentsline {subsection}{\numberline {51.8.11}\leavevmode {\color {Chapter }IsGroupHClass}}{803}{subsection.51.8.11}
\contentsline {subsection}{\numberline {51.8.12}\leavevmode {\color {Chapter }IsRegularDClass}}{803}{subsection.51.8.12}
\contentsline {subsection}{\numberline {51.8.13}\leavevmode {\color {Chapter }DisplaySemigroup}}{803}{subsection.51.8.13}
\contentsline {section}{\numberline {51.9}\leavevmode {\color {Chapter }Rees Matrix Semigroups}}{804}{section.51.9}
\contentsline {subsection}{\numberline {51.9.1}\leavevmode {\color {Chapter }ReesMatrixSemigroup}}{805}{subsection.51.9.1}
\contentsline {subsection}{\numberline {51.9.2}\leavevmode {\color {Chapter }ReesMatrixSubsemigroup}}{805}{subsection.51.9.2}
\contentsline {subsection}{\numberline {51.9.3}\leavevmode {\color {Chapter }IsomorphismReesMatrixSemigroup}}{806}{subsection.51.9.3}
\contentsline {subsection}{\numberline {51.9.4}\leavevmode {\color {Chapter }IsReesMatrixSemigroupElement}}{807}{subsection.51.9.4}
\contentsline {subsection}{\numberline {51.9.5}\leavevmode {\color {Chapter }ReesMatrixSemigroupElement}}{807}{subsection.51.9.5}
\contentsline {subsection}{\numberline {51.9.6}\leavevmode {\color {Chapter }IsReesMatrixSubsemigroup}}{808}{subsection.51.9.6}
\contentsline {subsection}{\numberline {51.9.7}\leavevmode {\color {Chapter }IsReesMatrixSemigroup}}{808}{subsection.51.9.7}
\contentsline {subsection}{\numberline {51.9.8}\leavevmode {\color {Chapter }Matrix}}{809}{subsection.51.9.8}
\contentsline {subsection}{\numberline {51.9.9}\leavevmode {\color {Chapter }Rows and columns}}{810}{subsection.51.9.9}
\contentsline {subsection}{\numberline {51.9.10}\leavevmode {\color {Chapter }UnderlyingSemigroup (for a Rees matrix semigroup)}}{810}{subsection.51.9.10}
\contentsline {subsection}{\numberline {51.9.11}\leavevmode {\color {Chapter }AssociatedReesMatrixSemigroupOfDClass}}{811}{subsection.51.9.11}
\contentsline {chapter}{\numberline {52}\leavevmode {\color {Chapter }Finitely Presented Semigroups and Monoids}}{812}{chapter.52}
\contentsline {section}{\numberline {52.1}\leavevmode {\color {Chapter }IsSubsemigroupFpSemigroup (Filter)}}{814}{section.52.1}
\contentsline {subsection}{\numberline {52.1.1}\leavevmode {\color {Chapter }IsSubsemigroupFpSemigroup}}{814}{subsection.52.1.1}
\contentsline {subsection}{\numberline {52.1.2}\leavevmode {\color {Chapter }IsFpSemigroup}}{814}{subsection.52.1.2}
\contentsline {subsection}{\numberline {52.1.3}\leavevmode {\color {Chapter }IsElementOfFpSemigroup}}{814}{subsection.52.1.3}
\contentsline {subsection}{\numberline {52.1.4}\leavevmode {\color {Chapter }FpGrpMonSmgOfFpGrpMonSmgElement}}{815}{subsection.52.1.4}
\contentsline {section}{\numberline {52.2}\leavevmode {\color {Chapter }Creating Finitely Presented Semigroups and Monoids}}{815}{section.52.2}
\contentsline {subsection}{\numberline {52.2.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }/ (for a free semigroup or monoid and a list of pairs of elements)}}{815}{subsection.52.2.1}
\contentsline {subsection}{\numberline {52.2.2}\leavevmode {\color {Chapter }FactorFreeSemigroupByRelations}}{815}{subsection.52.2.2}
\contentsline {subsection}{\numberline {52.2.3}\leavevmode {\color {Chapter }IsomorphismFpSemigroup}}{816}{subsection.52.2.3}
\contentsline {section}{\numberline {52.3}\leavevmode {\color {Chapter }Comparison of Elements of Finitely Presented Semigroups}}{816}{section.52.3}
\contentsline {subsection}{\numberline {52.3.1}\leavevmode {\color {Chapter }\texttt {\char 92\relax }= (for two elements in a f.p. semigroup)}}{816}{subsection.52.3.1}
\contentsline {section}{\numberline {52.4}\leavevmode {\color {Chapter }Preimages in the Free Semigroup or Monoid}}{816}{section.52.4}
\contentsline {subsection}{\numberline {52.4.1}\leavevmode {\color {Chapter }UnderlyingElement (of an element in a fp semigroup or monoid)}}{816}{subsection.52.4.1}
\contentsline {subsection}{\numberline {52.4.2}\leavevmode {\color {Chapter }ElementOfFpSemigroup}}{817}{subsection.52.4.2}
\contentsline {subsection}{\numberline {52.4.3}\leavevmode {\color {Chapter }FreeSemigroupOfFpSemigroup}}{817}{subsection.52.4.3}
\contentsline {subsection}{\numberline {52.4.4}\leavevmode {\color {Chapter }FreeGeneratorsOfFpSemigroup}}{817}{subsection.52.4.4}
\contentsline {subsection}{\numberline {52.4.5}\leavevmode {\color {Chapter }RelationsOfFpSemigroup}}{817}{subsection.52.4.5}
\contentsline {section}{\numberline {52.5}\leavevmode {\color {Chapter }Rewriting Systems and the Knuth-Bendix Procedure}}{818}{section.52.5}
\contentsline {subsection}{\numberline {52.5.1}\leavevmode {\color {Chapter }ReducedConfluentRewritingSystem}}{818}{subsection.52.5.1}
\contentsline {subsection}{\numberline {52.5.2}\leavevmode {\color {Chapter }KB{\textunderscore }REW}}{819}{subsection.52.5.2}
\contentsline {subsection}{\numberline {52.5.3}\leavevmode {\color {Chapter }KnuthBendixRewritingSystem}}{819}{subsection.52.5.3}
\contentsline {subsection}{\numberline {52.5.4}\leavevmode {\color {Chapter }SemigroupOfRewritingSystem}}{819}{subsection.52.5.4}
\contentsline {subsection}{\numberline {52.5.5}\leavevmode {\color {Chapter }FreeSemigroupOfRewritingSystem}}{820}{subsection.52.5.5}
\contentsline {section}{\numberline {52.6}\leavevmode {\color {Chapter }Todd-Coxeter Procedure}}{820}{section.52.6}
\contentsline {subsection}{\numberline {52.6.1}\leavevmode {\color {Chapter }CosetTableOfFpSemigroup}}{820}{subsection.52.6.1}
\contentsline {chapter}{\numberline {53}\leavevmode {\color {Chapter }Transformations}}{821}{chapter.53}
\contentsline {section}{\numberline {53.1}\leavevmode {\color {Chapter }The family and categories of transformations}}{822}{section.53.1}
\contentsline {subsection}{\numberline {53.1.1}\leavevmode {\color {Chapter }IsTransformation}}{822}{subsection.53.1.1}
\contentsline {subsection}{\numberline {53.1.2}\leavevmode {\color {Chapter }IsTransformationCollection}}{822}{subsection.53.1.2}
\contentsline {subsection}{\numberline {53.1.3}\leavevmode {\color {Chapter }TransformationFamily}}{823}{subsection.53.1.3}
\contentsline {section}{\numberline {53.2}\leavevmode {\color {Chapter }Creating transformations}}{823}{section.53.2}
\contentsline {subsection}{\numberline {53.2.1}\leavevmode {\color {Chapter }Transformation (for an image list)}}{823}{subsection.53.2.1}
\contentsline {subsection}{\numberline {53.2.2}\leavevmode {\color {Chapter }Transformation (for a source and destination)}}{823}{subsection.53.2.2}
\contentsline {subsection}{\numberline {53.2.3}\leavevmode {\color {Chapter }TransformationByImageAndKernel (for an image and kernel)}}{824}{subsection.53.2.3}
\contentsline {subsection}{\numberline {53.2.4}\leavevmode {\color {Chapter }Idempotent}}{824}{subsection.53.2.4}
\contentsline {subsection}{\numberline {53.2.5}\leavevmode {\color {Chapter }TransformationOp}}{824}{subsection.53.2.5}
\contentsline {subsection}{\numberline {53.2.6}\leavevmode {\color {Chapter }TransformationNumber}}{825}{subsection.53.2.6}
\contentsline {subsection}{\numberline {53.2.7}\leavevmode {\color {Chapter }RandomTransformation}}{826}{subsection.53.2.7}
\contentsline {subsection}{\numberline {53.2.8}\leavevmode {\color {Chapter }IdentityTransformation}}{826}{subsection.53.2.8}
\contentsline {subsection}{\numberline {53.2.9}\leavevmode {\color {Chapter }ConstantTransformation}}{826}{subsection.53.2.9}
\contentsline {section}{\numberline {53.3}\leavevmode {\color {Chapter }Changing the representation of a transformation}}{826}{section.53.3}
\contentsline {subsection}{\numberline {53.3.1}\leavevmode {\color {Chapter }AsTransformation}}{826}{subsection.53.3.1}
\contentsline {subsection}{\numberline {53.3.2}\leavevmode {\color {Chapter }RestrictedTransformation}}{828}{subsection.53.3.2}
\contentsline {subsection}{\numberline {53.3.3}\leavevmode {\color {Chapter }PermutationOfImage}}{828}{subsection.53.3.3}
\contentsline {section}{\numberline {53.4}\leavevmode {\color {Chapter }Operators for transformations}}{828}{section.53.4}
\contentsline {subsection}{\numberline {53.4.1}\leavevmode {\color {Chapter }PermLeftQuoTransformation}}{829}{subsection.53.4.1}
\contentsline {subsection}{\numberline {53.4.2}\leavevmode {\color {Chapter }IsInjectiveListTrans}}{829}{subsection.53.4.2}
\contentsline {subsection}{\numberline {53.4.3}\leavevmode {\color {Chapter }ComponentTransformationInt}}{830}{subsection.53.4.3}
\contentsline {subsection}{\numberline {53.4.4}\leavevmode {\color {Chapter }PreImagesOfTransformation}}{830}{subsection.53.4.4}
\contentsline {section}{\numberline {53.5}\leavevmode {\color {Chapter }Attributes for transformations}}{830}{section.53.5}
\contentsline {subsection}{\numberline {53.5.1}\leavevmode {\color {Chapter }DegreeOfTransformation}}{830}{subsection.53.5.1}
\contentsline {subsection}{\numberline {53.5.2}\leavevmode {\color {Chapter }ImageListOfTransformation}}{831}{subsection.53.5.2}
\contentsline {subsection}{\numberline {53.5.3}\leavevmode {\color {Chapter }ImageSetOfTransformation}}{831}{subsection.53.5.3}
\contentsline {subsection}{\numberline {53.5.4}\leavevmode {\color {Chapter }RankOfTransformation (for a transformation and a positive integer)}}{831}{subsection.53.5.4}
\contentsline {subsection}{\numberline {53.5.5}\leavevmode {\color {Chapter }MovedPoints (for a transformation)}}{832}{subsection.53.5.5}
\contentsline {subsection}{\numberline {53.5.6}\leavevmode {\color {Chapter }NrMovedPoints (for a transformation)}}{832}{subsection.53.5.6}
\contentsline {subsection}{\numberline {53.5.7}\leavevmode {\color {Chapter }SmallestMovedPoint (for a transformation)}}{833}{subsection.53.5.7}
\contentsline {subsection}{\numberline {53.5.8}\leavevmode {\color {Chapter }LargestMovedPoint (for a transformation)}}{833}{subsection.53.5.8}
\contentsline {subsection}{\numberline {53.5.9}\leavevmode {\color {Chapter }SmallestImageOfMovedPoint (for a transformation)}}{833}{subsection.53.5.9}
\contentsline {subsection}{\numberline {53.5.10}\leavevmode {\color {Chapter }LargestImageOfMovedPoint (for a transformation)}}{834}{subsection.53.5.10}
\contentsline {subsection}{\numberline {53.5.11}\leavevmode {\color {Chapter }FlatKernelOfTransformation}}{834}{subsection.53.5.11}
\contentsline {subsection}{\numberline {53.5.12}\leavevmode {\color {Chapter }KernelOfTransformation}}{835}{subsection.53.5.12}
\contentsline {subsection}{\numberline {53.5.13}\leavevmode {\color {Chapter }InverseOfTransformation}}{835}{subsection.53.5.13}
\contentsline {subsection}{\numberline {53.5.14}\leavevmode {\color {Chapter }Inverse (for a transformation)}}{835}{subsection.53.5.14}
\contentsline {subsection}{\numberline {53.5.15}\leavevmode {\color {Chapter }IndexPeriodOfTransformation}}{836}{subsection.53.5.15}
\contentsline {subsection}{\numberline {53.5.16}\leavevmode {\color {Chapter }SmallestIdempotentPower (for a transformation)}}{836}{subsection.53.5.16}
\contentsline {subsection}{\numberline {53.5.17}\leavevmode {\color {Chapter }ComponentsOfTransformation}}{836}{subsection.53.5.17}
\contentsline {subsection}{\numberline {53.5.18}\leavevmode {\color {Chapter }NrComponentsOfTransformation}}{837}{subsection.53.5.18}
\contentsline {subsection}{\numberline {53.5.19}\leavevmode {\color {Chapter }ComponentRepsOfTransformation}}{837}{subsection.53.5.19}
\contentsline {subsection}{\numberline {53.5.20}\leavevmode {\color {Chapter }CyclesOfTransformation}}{838}{subsection.53.5.20}
\contentsline {subsection}{\numberline {53.5.21}\leavevmode {\color {Chapter }CycleTransformationInt}}{838}{subsection.53.5.21}
\contentsline {subsection}{\numberline {53.5.22}\leavevmode {\color {Chapter }LeftOne (for a transformation)}}{838}{subsection.53.5.22}
\contentsline {subsection}{\numberline {53.5.23}\leavevmode {\color {Chapter }TrimTransformation}}{839}{subsection.53.5.23}
\contentsline {section}{\numberline {53.6}\leavevmode {\color {Chapter }Displaying transformations}}{840}{section.53.6}
\contentsline {section}{\numberline {53.7}\leavevmode {\color {Chapter }Semigroups of transformations}}{840}{section.53.7}
\contentsline {subsection}{\numberline {53.7.1}\leavevmode {\color {Chapter }IsTransformationSemigroup}}{841}{subsection.53.7.1}
\contentsline {subsection}{\numberline {53.7.2}\leavevmode {\color {Chapter }DegreeOfTransformationSemigroup}}{841}{subsection.53.7.2}
\contentsline {subsection}{\numberline {53.7.3}\leavevmode {\color {Chapter }FullTransformationSemigroup}}{842}{subsection.53.7.3}
\contentsline {subsection}{\numberline {53.7.4}\leavevmode {\color {Chapter }IsFullTransformationSemigroup}}{842}{subsection.53.7.4}
\contentsline {subsection}{\numberline {53.7.5}\leavevmode {\color {Chapter }IsomorphismTransformationSemigroup}}{842}{subsection.53.7.5}
\contentsline {subsection}{\numberline {53.7.6}\leavevmode {\color {Chapter }AntiIsomorphismTransformationSemigroup}}{843}{subsection.53.7.6}
\contentsline {chapter}{\numberline {54}\leavevmode {\color {Chapter }Partial permutations}}{845}{chapter.54}
\contentsline {section}{\numberline {54.1}\leavevmode {\color {Chapter }The family and categories of partial permutations}}{847}{section.54.1}
\contentsline {subsection}{\numberline {54.1.1}\leavevmode {\color {Chapter }IsPartialPerm}}{847}{subsection.54.1.1}
\contentsline {subsection}{\numberline {54.1.2}\leavevmode {\color {Chapter }IsPartialPermCollection}}{847}{subsection.54.1.2}
\contentsline {subsection}{\numberline {54.1.3}\leavevmode {\color {Chapter }PartialPermFamily}}{847}{subsection.54.1.3}
\contentsline {section}{\numberline {54.2}\leavevmode {\color {Chapter }Creating partial permutations}}{847}{section.54.2}
\contentsline {subsection}{\numberline {54.2.1}\leavevmode {\color {Chapter }PartialPerm (for a domain and image)}}{847}{subsection.54.2.1}
\contentsline {subsection}{\numberline {54.2.2}\leavevmode {\color {Chapter }PartialPermOp}}{848}{subsection.54.2.2}
\contentsline {subsection}{\numberline {54.2.3}\leavevmode {\color {Chapter }RestrictedPartialPerm}}{849}{subsection.54.2.3}
\contentsline {subsection}{\numberline {54.2.4}\leavevmode {\color {Chapter }JoinOfPartialPerms}}{849}{subsection.54.2.4}
\contentsline {subsection}{\numberline {54.2.5}\leavevmode {\color {Chapter }MeetOfPartialPerms}}{850}{subsection.54.2.5}
\contentsline {subsection}{\numberline {54.2.6}\leavevmode {\color {Chapter }EmptyPartialPerm}}{850}{subsection.54.2.6}
\contentsline {subsection}{\numberline {54.2.7}\leavevmode {\color {Chapter }RandomPartialPerm}}{851}{subsection.54.2.7}
\contentsline {section}{\numberline {54.3}\leavevmode {\color {Chapter }Attributes for partial permutations}}{851}{section.54.3}
\contentsline {subsection}{\numberline {54.3.1}\leavevmode {\color {Chapter }DegreeOfPartialPerm}}{851}{subsection.54.3.1}
\contentsline {subsection}{\numberline {54.3.2}\leavevmode {\color {Chapter }CodegreeOfPartialPerm}}{852}{subsection.54.3.2}
\contentsline {subsection}{\numberline {54.3.3}\leavevmode {\color {Chapter }RankOfPartialPerm}}{852}{subsection.54.3.3}
\contentsline {subsection}{\numberline {54.3.4}\leavevmode {\color {Chapter }DomainOfPartialPerm}}{852}{subsection.54.3.4}
\contentsline {subsection}{\numberline {54.3.5}\leavevmode {\color {Chapter }ImageOfPartialPermCollection}}{852}{subsection.54.3.5}
\contentsline {subsection}{\numberline {54.3.6}\leavevmode {\color {Chapter }ImageListOfPartialPerm}}{853}{subsection.54.3.6}
\contentsline {subsection}{\numberline {54.3.7}\leavevmode {\color {Chapter }ImageSetOfPartialPerm}}{853}{subsection.54.3.7}
\contentsline {subsection}{\numberline {54.3.8}\leavevmode {\color {Chapter }FixedPointsOfPartialPerm (for a partial perm)}}{853}{subsection.54.3.8}
\contentsline {subsection}{\numberline {54.3.9}\leavevmode {\color {Chapter }MovedPoints (for a partial perm)}}{854}{subsection.54.3.9}
\contentsline {subsection}{\numberline {54.3.10}\leavevmode {\color {Chapter }NrFixedPoints (for a partial perm)}}{854}{subsection.54.3.10}
\contentsline {subsection}{\numberline {54.3.11}\leavevmode {\color {Chapter }NrMovedPoints (for a partial perm)}}{854}{subsection.54.3.11}
\contentsline {subsection}{\numberline {54.3.12}\leavevmode {\color {Chapter }SmallestMovedPoint (for a partial perm)}}{855}{subsection.54.3.12}
\contentsline {subsection}{\numberline {54.3.13}\leavevmode {\color {Chapter }LargestMovedPoint (for a partial perm)}}{855}{subsection.54.3.13}
\contentsline {subsection}{\numberline {54.3.14}\leavevmode {\color {Chapter }SmallestImageOfMovedPoint (for a partial permutation)}}{855}{subsection.54.3.14}
\contentsline {subsection}{\numberline {54.3.15}\leavevmode {\color {Chapter }LargestImageOfMovedPoint (for a partial permutation)}}{856}{subsection.54.3.15}
\contentsline {subsection}{\numberline {54.3.16}\leavevmode {\color {Chapter }IndexPeriodOfPartialPerm}}{856}{subsection.54.3.16}
\contentsline {subsection}{\numberline {54.3.17}\leavevmode {\color {Chapter }SmallestIdempotentPower (for a partial perm)}}{857}{subsection.54.3.17}
\contentsline {subsection}{\numberline {54.3.18}\leavevmode {\color {Chapter }ComponentsOfPartialPerm}}{857}{subsection.54.3.18}
\contentsline {subsection}{\numberline {54.3.19}\leavevmode {\color {Chapter }NrComponentsOfPartialPerm}}{857}{subsection.54.3.19}
\contentsline {subsection}{\numberline {54.3.20}\leavevmode {\color {Chapter }ComponentRepsOfPartialPerm}}{858}{subsection.54.3.20}
\contentsline {subsection}{\numberline {54.3.21}\leavevmode {\color {Chapter }LeftOne (for a partial perm)}}{858}{subsection.54.3.21}
\contentsline {subsection}{\numberline {54.3.22}\leavevmode {\color {Chapter }One (for a partial perm)}}{858}{subsection.54.3.22}
\contentsline {subsection}{\numberline {54.3.23}\leavevmode {\color {Chapter }MultiplicativeZero (for a partial perm)}}{859}{subsection.54.3.23}
\contentsline {section}{\numberline {54.4}\leavevmode {\color {Chapter }Changing the representation of a partial permutation}}{859}{section.54.4}
\contentsline {subsection}{\numberline {54.4.1}\leavevmode {\color {Chapter }AsPartialPerm (for a permutation and a set of positive integers)}}{859}{subsection.54.4.1}
\contentsline {subsection}{\numberline {54.4.2}\leavevmode {\color {Chapter }AsPartialPerm (for a transformation and a set of positive integer)}}{860}{subsection.54.4.2}
\contentsline {section}{\numberline {54.5}\leavevmode {\color {Chapter }Operators and operations for partial permutations}}{860}{section.54.5}
\contentsline {subsection}{\numberline {54.5.1}\leavevmode {\color {Chapter }PermLeftQuoPartialPerm}}{861}{subsection.54.5.1}
\contentsline {subsection}{\numberline {54.5.2}\leavevmode {\color {Chapter }PreImagePartialPerm}}{861}{subsection.54.5.2}
\contentsline {subsection}{\numberline {54.5.3}\leavevmode {\color {Chapter }ComponentPartialPermInt}}{862}{subsection.54.5.3}
\contentsline {subsection}{\numberline {54.5.4}\leavevmode {\color {Chapter }NaturalLeqPartialPerm}}{862}{subsection.54.5.4}
\contentsline {subsection}{\numberline {54.5.5}\leavevmode {\color {Chapter }ShortLexLeqPartialPerm}}{863}{subsection.54.5.5}
\contentsline {subsection}{\numberline {54.5.6}\leavevmode {\color {Chapter }TrimPartialPerm}}{863}{subsection.54.5.6}
\contentsline {section}{\numberline {54.6}\leavevmode {\color {Chapter }Displaying partial permutations}}{864}{section.54.6}
\contentsline {section}{\numberline {54.7}\leavevmode {\color {Chapter }Semigroups and inverse semigroups of partial permutations}}{865}{section.54.7}
\contentsline {subsection}{\numberline {54.7.1}\leavevmode {\color {Chapter }IsPartialPermSemigroup}}{866}{subsection.54.7.1}
\contentsline {subsection}{\numberline {54.7.2}\leavevmode {\color {Chapter }DegreeOfPartialPermSemigroup}}{866}{subsection.54.7.2}
\contentsline {subsection}{\numberline {54.7.3}\leavevmode {\color {Chapter }SymmetricInverseSemigroup}}{867}{subsection.54.7.3}
\contentsline {subsection}{\numberline {54.7.4}\leavevmode {\color {Chapter }IsSymmetricInverseSemigroup}}{867}{subsection.54.7.4}
\contentsline {subsection}{\numberline {54.7.5}\leavevmode {\color {Chapter }NaturalPartialOrder}}{867}{subsection.54.7.5}
\contentsline {subsection}{\numberline {54.7.6}\leavevmode {\color {Chapter }IsomorphismPartialPermSemigroup}}{868}{subsection.54.7.6}
\contentsline {chapter}{\numberline {55}\leavevmode {\color {Chapter }Additive Magmas}}{870}{chapter.55}
\contentsline {section}{\numberline {55.1}\leavevmode {\color {Chapter }(Near-)Additive Magma Categories}}{870}{section.55.1}
\contentsline {subsection}{\numberline {55.1.1}\leavevmode {\color {Chapter }IsNearAdditiveMagma}}{870}{subsection.55.1.1}
\contentsline {subsection}{\numberline {55.1.2}\leavevmode {\color {Chapter }IsNearAdditiveMagmaWithZero}}{870}{subsection.55.1.2}
\contentsline {subsection}{\numberline {55.1.3}\leavevmode {\color {Chapter }IsNearAdditiveGroup}}{870}{subsection.55.1.3}
\contentsline {subsection}{\numberline {55.1.4}\leavevmode {\color {Chapter }IsAdditiveMagma}}{871}{subsection.55.1.4}
\contentsline {subsection}{\numberline {55.1.5}\leavevmode {\color {Chapter }IsAdditiveMagmaWithZero}}{871}{subsection.55.1.5}
\contentsline {subsection}{\numberline {55.1.6}\leavevmode {\color {Chapter }IsAdditiveGroup}}{871}{subsection.55.1.6}
\contentsline {section}{\numberline {55.2}\leavevmode {\color {Chapter }(Near-)Additive Magma Generation}}{872}{section.55.2}
\contentsline {subsection}{\numberline {55.2.1}\leavevmode {\color {Chapter }NearAdditiveMagma}}{872}{subsection.55.2.1}
\contentsline {subsection}{\numberline {55.2.2}\leavevmode {\color {Chapter }NearAdditiveMagmaWithZero}}{872}{subsection.55.2.2}
\contentsline {subsection}{\numberline {55.2.3}\leavevmode {\color {Chapter }NearAdditiveGroup}}{872}{subsection.55.2.3}
\contentsline {subsection}{\numberline {55.2.4}\leavevmode {\color {Chapter }NearAdditiveMagmaByGenerators}}{872}{subsection.55.2.4}
\contentsline {subsection}{\numberline {55.2.5}\leavevmode {\color {Chapter }NearAdditiveMagmaWithZeroByGenerators}}{872}{subsection.55.2.5}
\contentsline {subsection}{\numberline {55.2.6}\leavevmode {\color {Chapter }NearAdditiveGroupByGenerators}}{873}{subsection.55.2.6}
\contentsline {subsection}{\numberline {55.2.7}\leavevmode {\color {Chapter }SubnearAdditiveMagma}}{873}{subsection.55.2.7}
\contentsline {subsection}{\numberline {55.2.8}\leavevmode {\color {Chapter }SubnearAdditiveMagmaWithZero}}{873}{subsection.55.2.8}
\contentsline {subsection}{\numberline {55.2.9}\leavevmode {\color {Chapter }SubnearAdditiveGroup}}{873}{subsection.55.2.9}
\contentsline {section}{\numberline {55.3}\leavevmode {\color {Chapter }Attributes and Properties for (Near-)Additive Magmas}}{874}{section.55.3}
\contentsline {subsection}{\numberline {55.3.1}\leavevmode {\color {Chapter }IsAdditivelyCommutative}}{874}{subsection.55.3.1}
\contentsline {subsection}{\numberline {55.3.2}\leavevmode {\color {Chapter }GeneratorsOfNearAdditiveMagma}}{874}{subsection.55.3.2}
\contentsline {subsection}{\numberline {55.3.3}\leavevmode {\color {Chapter }GeneratorsOfNearAdditiveMagmaWithZero}}{874}{subsection.55.3.3}
\contentsline {subsection}{\numberline {55.3.4}\leavevmode {\color {Chapter }GeneratorsOfNearAdditiveGroup}}{874}{subsection.55.3.4}
\contentsline {subsection}{\numberline {55.3.5}\leavevmode {\color {Chapter }AdditiveNeutralElement}}{874}{subsection.55.3.5}
\contentsline {subsection}{\numberline {55.3.6}\leavevmode {\color {Chapter }TrivialSubnearAdditiveMagmaWithZero}}{875}{subsection.55.3.6}
\contentsline {section}{\numberline {55.4}\leavevmode {\color {Chapter }Operations for (Near-)Additive Magmas}}{875}{section.55.4}
\contentsline {subsection}{\numberline {55.4.1}\leavevmode {\color {Chapter }ClosureNearAdditiveGroup}}{875}{subsection.55.4.1}
\contentsline {subsection}{\numberline {55.4.2}\leavevmode {\color {Chapter }ShowAdditionTable}}{875}{subsection.55.4.2}
\contentsline {chapter}{\numberline {56}\leavevmode {\color {Chapter }Rings}}{876}{chapter.56}
\contentsline {section}{\numberline {56.1}\leavevmode {\color {Chapter }Generating Rings}}{876}{section.56.1}
\contentsline {subsection}{\numberline {56.1.1}\leavevmode {\color {Chapter }IsRing}}{876}{subsection.56.1.1}
\contentsline {subsection}{\numberline {56.1.2}\leavevmode {\color {Chapter }Ring}}{876}{subsection.56.1.2}
\contentsline {subsection}{\numberline {56.1.3}\leavevmode {\color {Chapter }DefaultRing}}{877}{subsection.56.1.3}
\contentsline {subsection}{\numberline {56.1.4}\leavevmode {\color {Chapter }RingByGenerators}}{877}{subsection.56.1.4}
\contentsline {subsection}{\numberline {56.1.5}\leavevmode {\color {Chapter }DefaultRingByGenerators}}{877}{subsection.56.1.5}
\contentsline {subsection}{\numberline {56.1.6}\leavevmode {\color {Chapter }GeneratorsOfRing}}{878}{subsection.56.1.6}
\contentsline {subsection}{\numberline {56.1.7}\leavevmode {\color {Chapter }Subring}}{878}{subsection.56.1.7}
\contentsline {subsection}{\numberline {56.1.8}\leavevmode {\color {Chapter }ClosureRing}}{878}{subsection.56.1.8}
\contentsline {subsection}{\numberline {56.1.9}\leavevmode {\color {Chapter }Quotient}}{878}{subsection.56.1.9}
\contentsline {section}{\numberline {56.2}\leavevmode {\color {Chapter }Ideals of Rings}}{879}{section.56.2}
\contentsline {subsection}{\numberline {56.2.1}\leavevmode {\color {Chapter }TwoSidedIdeal}}{879}{subsection.56.2.1}
\contentsline {subsection}{\numberline {56.2.2}\leavevmode {\color {Chapter }TwoSidedIdealNC}}{880}{subsection.56.2.2}
\contentsline {subsection}{\numberline {56.2.3}\leavevmode {\color {Chapter }IsTwoSidedIdeal}}{880}{subsection.56.2.3}
\contentsline {subsection}{\numberline {56.2.4}\leavevmode {\color {Chapter }TwoSidedIdealByGenerators}}{881}{subsection.56.2.4}
\contentsline {subsection}{\numberline {56.2.5}\leavevmode {\color {Chapter }LeftIdealByGenerators}}{881}{subsection.56.2.5}
\contentsline {subsection}{\numberline {56.2.6}\leavevmode {\color {Chapter }RightIdealByGenerators}}{881}{subsection.56.2.6}
\contentsline {subsection}{\numberline {56.2.7}\leavevmode {\color {Chapter }GeneratorsOfTwoSidedIdeal}}{881}{subsection.56.2.7}
\contentsline {subsection}{\numberline {56.2.8}\leavevmode {\color {Chapter }GeneratorsOfLeftIdeal}}{881}{subsection.56.2.8}
\contentsline {subsection}{\numberline {56.2.9}\leavevmode {\color {Chapter }GeneratorsOfRightIdeal}}{882}{subsection.56.2.9}
\contentsline {subsection}{\numberline {56.2.10}\leavevmode {\color {Chapter }LeftActingRingOfIdeal}}{882}{subsection.56.2.10}
\contentsline {subsection}{\numberline {56.2.11}\leavevmode {\color {Chapter }AsLeftIdeal}}{882}{subsection.56.2.11}
\contentsline {section}{\numberline {56.3}\leavevmode {\color {Chapter }Rings With One}}{882}{section.56.3}
\contentsline {subsection}{\numberline {56.3.1}\leavevmode {\color {Chapter }IsRingWithOne}}{882}{subsection.56.3.1}
\contentsline {subsection}{\numberline {56.3.2}\leavevmode {\color {Chapter }RingWithOne}}{883}{subsection.56.3.2}
\contentsline {subsection}{\numberline {56.3.3}\leavevmode {\color {Chapter }RingWithOneByGenerators}}{883}{subsection.56.3.3}
\contentsline {subsection}{\numberline {56.3.4}\leavevmode {\color {Chapter }GeneratorsOfRingWithOne}}{883}{subsection.56.3.4}
\contentsline {subsection}{\numberline {56.3.5}\leavevmode {\color {Chapter }SubringWithOne}}{883}{subsection.56.3.5}
\contentsline {section}{\numberline {56.4}\leavevmode {\color {Chapter }Properties of Rings}}{884}{section.56.4}
\contentsline {subsection}{\numberline {56.4.1}\leavevmode {\color {Chapter }IsIntegralRing}}{884}{subsection.56.4.1}
\contentsline {subsection}{\numberline {56.4.2}\leavevmode {\color {Chapter }IsUniqueFactorizationRing}}{884}{subsection.56.4.2}
\contentsline {subsection}{\numberline {56.4.3}\leavevmode {\color {Chapter }IsLDistributive}}{884}{subsection.56.4.3}
\contentsline {subsection}{\numberline {56.4.4}\leavevmode {\color {Chapter }IsRDistributive}}{885}{subsection.56.4.4}
\contentsline {subsection}{\numberline {56.4.5}\leavevmode {\color {Chapter }IsDistributive}}{885}{subsection.56.4.5}
\contentsline {subsection}{\numberline {56.4.6}\leavevmode {\color {Chapter }IsAnticommutative}}{885}{subsection.56.4.6}
\contentsline {subsection}{\numberline {56.4.7}\leavevmode {\color {Chapter }IsZeroSquaredRing}}{885}{subsection.56.4.7}
\contentsline {subsection}{\numberline {56.4.8}\leavevmode {\color {Chapter }IsJacobianRing}}{885}{subsection.56.4.8}
\contentsline {section}{\numberline {56.5}\leavevmode {\color {Chapter }Units and Factorizations}}{886}{section.56.5}
\contentsline {subsection}{\numberline {56.5.1}\leavevmode {\color {Chapter }IsUnit}}{886}{subsection.56.5.1}
\contentsline {subsection}{\numberline {56.5.2}\leavevmode {\color {Chapter }Units}}{886}{subsection.56.5.2}
\contentsline {subsection}{\numberline {56.5.3}\leavevmode {\color {Chapter }IsAssociated}}{886}{subsection.56.5.3}
\contentsline {subsection}{\numberline {56.5.4}\leavevmode {\color {Chapter }Associates}}{886}{subsection.56.5.4}
\contentsline {subsection}{\numberline {56.5.5}\leavevmode {\color {Chapter }StandardAssociate}}{887}{subsection.56.5.5}
\contentsline {subsection}{\numberline {56.5.6}\leavevmode {\color {Chapter }StandardAssociateUnit}}{887}{subsection.56.5.6}
\contentsline {subsection}{\numberline {56.5.7}\leavevmode {\color {Chapter }IsIrreducibleRingElement}}{887}{subsection.56.5.7}
\contentsline {subsection}{\numberline {56.5.8}\leavevmode {\color {Chapter }IsPrime}}{887}{subsection.56.5.8}
\contentsline {subsection}{\numberline {56.5.9}\leavevmode {\color {Chapter }Factors}}{888}{subsection.56.5.9}
\contentsline {subsection}{\numberline {56.5.10}\leavevmode {\color {Chapter }PadicValuation}}{888}{subsection.56.5.10}
\contentsline {section}{\numberline {56.6}\leavevmode {\color {Chapter }Euclidean Rings}}{888}{section.56.6}
\contentsline {subsection}{\numberline {56.6.1}\leavevmode {\color {Chapter }IsEuclideanRing}}{888}{subsection.56.6.1}
\contentsline {subsection}{\numberline {56.6.2}\leavevmode {\color {Chapter }EuclideanDegree}}{889}{subsection.56.6.2}
\contentsline {subsection}{\numberline {56.6.3}\leavevmode {\color {Chapter }EuclideanQuotient}}{889}{subsection.56.6.3}
\contentsline {subsection}{\numberline {56.6.4}\leavevmode {\color {Chapter }EuclideanRemainder}}{889}{subsection.56.6.4}
\contentsline {subsection}{\numberline {56.6.5}\leavevmode {\color {Chapter }QuotientRemainder}}{889}{subsection.56.6.5}
\contentsline {section}{\numberline {56.7}\leavevmode {\color {Chapter }Gcd and Lcm}}{890}{section.56.7}
\contentsline {subsection}{\numberline {56.7.1}\leavevmode {\color {Chapter }Gcd}}{890}{subsection.56.7.1}
\contentsline {subsection}{\numberline {56.7.2}\leavevmode {\color {Chapter }GcdOp}}{890}{subsection.56.7.2}
\contentsline {subsection}{\numberline {56.7.3}\leavevmode {\color {Chapter }GcdRepresentation}}{890}{subsection.56.7.3}
\contentsline {subsection}{\numberline {56.7.4}\leavevmode {\color {Chapter }GcdRepresentationOp}}{891}{subsection.56.7.4}
\contentsline {subsection}{\numberline {56.7.5}\leavevmode {\color {Chapter }ShowGcd}}{891}{subsection.56.7.5}
\contentsline {subsection}{\numberline {56.7.6}\leavevmode {\color {Chapter }Lcm}}{891}{subsection.56.7.6}
\contentsline {subsection}{\numberline {56.7.7}\leavevmode {\color {Chapter }LcmOp}}{891}{subsection.56.7.7}
\contentsline {subsection}{\numberline {56.7.8}\leavevmode {\color {Chapter }QuotientMod}}{892}{subsection.56.7.8}
\contentsline {subsection}{\numberline {56.7.9}\leavevmode {\color {Chapter }PowerMod}}{892}{subsection.56.7.9}
\contentsline {subsection}{\numberline {56.7.10}\leavevmode {\color {Chapter }InterpolatedPolynomial}}{892}{subsection.56.7.10}
\contentsline {section}{\numberline {56.8}\leavevmode {\color {Chapter }Homomorphisms of Rings}}{892}{section.56.8}
\contentsline {subsection}{\numberline {56.8.1}\leavevmode {\color {Chapter }RingGeneralMappingByImages}}{893}{subsection.56.8.1}
\contentsline {subsection}{\numberline {56.8.2}\leavevmode {\color {Chapter }RingHomomorphismByImages}}{893}{subsection.56.8.2}
\contentsline {subsection}{\numberline {56.8.3}\leavevmode {\color {Chapter }RingHomomorphismByImagesNC}}{893}{subsection.56.8.3}
\contentsline {subsection}{\numberline {56.8.4}\leavevmode {\color {Chapter }NaturalHomomorphismByIdeal}}{893}{subsection.56.8.4}
\contentsline {section}{\numberline {56.9}\leavevmode {\color {Chapter }Small Rings}}{893}{section.56.9}
\contentsline {subsection}{\numberline {56.9.1}\leavevmode {\color {Chapter }SmallRing}}{894}{subsection.56.9.1}
\contentsline {subsection}{\numberline {56.9.2}\leavevmode {\color {Chapter }NumberSmallRings}}{894}{subsection.56.9.2}
\contentsline {subsection}{\numberline {56.9.3}\leavevmode {\color {Chapter }Subrings}}{894}{subsection.56.9.3}
\contentsline {subsection}{\numberline {56.9.4}\leavevmode {\color {Chapter }Ideals}}{894}{subsection.56.9.4}
\contentsline {subsection}{\numberline {56.9.5}\leavevmode {\color {Chapter }DirectSum}}{895}{subsection.56.9.5}
\contentsline {subsection}{\numberline {56.9.6}\leavevmode {\color {Chapter }RingByStructureConstants}}{895}{subsection.56.9.6}
\contentsline {chapter}{\numberline {57}\leavevmode {\color {Chapter }Modules}}{896}{chapter.57}
\contentsline {section}{\numberline {57.1}\leavevmode {\color {Chapter }Generating modules}}{896}{section.57.1}
\contentsline {subsection}{\numberline {57.1.1}\leavevmode {\color {Chapter }IsLeftOperatorAdditiveGroup}}{896}{subsection.57.1.1}
\contentsline {subsection}{\numberline {57.1.2}\leavevmode {\color {Chapter }IsLeftModule}}{896}{subsection.57.1.2}
\contentsline {subsection}{\numberline {57.1.3}\leavevmode {\color {Chapter }GeneratorsOfLeftOperatorAdditiveGroup}}{896}{subsection.57.1.3}
\contentsline {subsection}{\numberline {57.1.4}\leavevmode {\color {Chapter }GeneratorsOfLeftModule}}{896}{subsection.57.1.4}
\contentsline {subsection}{\numberline {57.1.5}\leavevmode {\color {Chapter }AsLeftModule}}{897}{subsection.57.1.5}
\contentsline {subsection}{\numberline {57.1.6}\leavevmode {\color {Chapter }IsRightOperatorAdditiveGroup}}{897}{subsection.57.1.6}
\contentsline {subsection}{\numberline {57.1.7}\leavevmode {\color {Chapter }IsRightModule}}{897}{subsection.57.1.7}
\contentsline {subsection}{\numberline {57.1.8}\leavevmode {\color {Chapter }GeneratorsOfRightOperatorAdditiveGroup}}{897}{subsection.57.1.8}
\contentsline {subsection}{\numberline {57.1.9}\leavevmode {\color {Chapter }GeneratorsOfRightModule}}{897}{subsection.57.1.9}
\contentsline {subsection}{\numberline {57.1.10}\leavevmode {\color {Chapter }LeftModuleByGenerators}}{898}{subsection.57.1.10}
\contentsline {subsection}{\numberline {57.1.11}\leavevmode {\color {Chapter }LeftActingDomain}}{898}{subsection.57.1.11}
\contentsline {section}{\numberline {57.2}\leavevmode {\color {Chapter }Submodules}}{898}{section.57.2}
\contentsline {subsection}{\numberline {57.2.1}\leavevmode {\color {Chapter }Submodule}}{898}{subsection.57.2.1}
\contentsline {subsection}{\numberline {57.2.2}\leavevmode {\color {Chapter }SubmoduleNC}}{898}{subsection.57.2.2}
\contentsline {subsection}{\numberline {57.2.3}\leavevmode {\color {Chapter }ClosureLeftModule}}{898}{subsection.57.2.3}
\contentsline {subsection}{\numberline {57.2.4}\leavevmode {\color {Chapter }TrivialSubmodule}}{899}{subsection.57.2.4}
\contentsline {section}{\numberline {57.3}\leavevmode {\color {Chapter }Free Modules}}{899}{section.57.3}
\contentsline {subsection}{\numberline {57.3.1}\leavevmode {\color {Chapter }IsFreeLeftModule}}{899}{subsection.57.3.1}
\contentsline {subsection}{\numberline {57.3.2}\leavevmode {\color {Chapter }FreeLeftModule}}{899}{subsection.57.3.2}
\contentsline {subsection}{\numberline {57.3.3}\leavevmode {\color {Chapter }Dimension}}{900}{subsection.57.3.3}
\contentsline {subsection}{\numberline {57.3.4}\leavevmode {\color {Chapter }IsFiniteDimensional}}{900}{subsection.57.3.4}
\contentsline {subsection}{\numberline {57.3.5}\leavevmode {\color {Chapter }UseBasis}}{900}{subsection.57.3.5}
\contentsline {subsection}{\numberline {57.3.6}\leavevmode {\color {Chapter }IsRowModule}}{900}{subsection.57.3.6}
\contentsline {subsection}{\numberline {57.3.7}\leavevmode {\color {Chapter }IsMatrixModule}}{901}{subsection.57.3.7}
\contentsline {subsection}{\numberline {57.3.8}\leavevmode {\color {Chapter }IsFullRowModule}}{901}{subsection.57.3.8}
\contentsline {subsection}{\numberline {57.3.9}\leavevmode {\color {Chapter }FullRowModule}}{901}{subsection.57.3.9}
\contentsline {subsection}{\numberline {57.3.10}\leavevmode {\color {Chapter }IsFullMatrixModule}}{901}{subsection.57.3.10}
\contentsline {subsection}{\numberline {57.3.11}\leavevmode {\color {Chapter }FullMatrixModule}}{901}{subsection.57.3.11}
\contentsline {chapter}{\numberline {58}\leavevmode {\color {Chapter }Fields and Division Rings}}{902}{chapter.58}
\contentsline {section}{\numberline {58.1}\leavevmode {\color {Chapter }Generating Fields}}{902}{section.58.1}
\contentsline {subsection}{\numberline {58.1.1}\leavevmode {\color {Chapter }IsDivisionRing}}{902}{subsection.58.1.1}
\contentsline {subsection}{\numberline {58.1.2}\leavevmode {\color {Chapter }IsField}}{902}{subsection.58.1.2}
\contentsline {subsection}{\numberline {58.1.3}\leavevmode {\color {Chapter }Field (for several generators)}}{903}{subsection.58.1.3}
\contentsline {subsection}{\numberline {58.1.4}\leavevmode {\color {Chapter }DefaultField (for several generators)}}{903}{subsection.58.1.4}
\contentsline {subsection}{\numberline {58.1.5}\leavevmode {\color {Chapter }DefaultFieldByGenerators}}{903}{subsection.58.1.5}
\contentsline {subsection}{\numberline {58.1.6}\leavevmode {\color {Chapter }GeneratorsOfDivisionRing}}{904}{subsection.58.1.6}
\contentsline {subsection}{\numberline {58.1.7}\leavevmode {\color {Chapter }GeneratorsOfField}}{904}{subsection.58.1.7}
\contentsline {subsection}{\numberline {58.1.8}\leavevmode {\color {Chapter }DivisionRingByGenerators}}{904}{subsection.58.1.8}
\contentsline {subsection}{\numberline {58.1.9}\leavevmode {\color {Chapter }AsDivisionRing}}{904}{subsection.58.1.9}
\contentsline {section}{\numberline {58.2}\leavevmode {\color {Chapter }Subfields of Fields}}{904}{section.58.2}
\contentsline {subsection}{\numberline {58.2.1}\leavevmode {\color {Chapter }Subfield}}{904}{subsection.58.2.1}
\contentsline {subsection}{\numberline {58.2.2}\leavevmode {\color {Chapter }FieldOverItselfByGenerators}}{905}{subsection.58.2.2}
\contentsline {subsection}{\numberline {58.2.3}\leavevmode {\color {Chapter }PrimitiveElement}}{905}{subsection.58.2.3}
\contentsline {subsection}{\numberline {58.2.4}\leavevmode {\color {Chapter }PrimeField}}{905}{subsection.58.2.4}
\contentsline {subsection}{\numberline {58.2.5}\leavevmode {\color {Chapter }IsPrimeField}}{905}{subsection.58.2.5}
\contentsline {subsection}{\numberline {58.2.6}\leavevmode {\color {Chapter }DegreeOverPrimeField}}{905}{subsection.58.2.6}
\contentsline {subsection}{\numberline {58.2.7}\leavevmode {\color {Chapter }DefiningPolynomial}}{905}{subsection.58.2.7}
\contentsline {subsection}{\numberline {58.2.8}\leavevmode {\color {Chapter }RootOfDefiningPolynomial}}{905}{subsection.58.2.8}
\contentsline {subsection}{\numberline {58.2.9}\leavevmode {\color {Chapter }FieldExtension}}{906}{subsection.58.2.9}
\contentsline {subsection}{\numberline {58.2.10}\leavevmode {\color {Chapter }Subfields}}{906}{subsection.58.2.10}
\contentsline {section}{\numberline {58.3}\leavevmode {\color {Chapter }Galois Action}}{906}{section.58.3}
\contentsline {subsection}{\numberline {58.3.1}\leavevmode {\color {Chapter }GaloisGroup (of field)}}{907}{subsection.58.3.1}
\contentsline {subsection}{\numberline {58.3.2}\leavevmode {\color {Chapter }MinimalPolynomial (over a field)}}{907}{subsection.58.3.2}
\contentsline {subsection}{\numberline {58.3.3}\leavevmode {\color {Chapter }TracePolynomial}}{907}{subsection.58.3.3}
\contentsline {subsection}{\numberline {58.3.4}\leavevmode {\color {Chapter }Norm}}{908}{subsection.58.3.4}
\contentsline {subsection}{\numberline {58.3.5}\leavevmode {\color {Chapter }Traces of field elements and matrices}}{908}{subsection.58.3.5}
\contentsline {subsection}{\numberline {58.3.6}\leavevmode {\color {Chapter }Conjugates}}{908}{subsection.58.3.6}
\contentsline {subsection}{\numberline {58.3.7}\leavevmode {\color {Chapter }NormalBase}}{909}{subsection.58.3.7}
\contentsline {chapter}{\numberline {59}\leavevmode {\color {Chapter }Finite Fields}}{910}{chapter.59}
\contentsline {section}{\numberline {59.1}\leavevmode {\color {Chapter }Finite Field Elements}}{910}{section.59.1}
\contentsline {subsection}{\numberline {59.1.1}\leavevmode {\color {Chapter }IsFFE}}{910}{subsection.59.1.1}
\contentsline {subsection}{\numberline {59.1.2}\leavevmode {\color {Chapter }Z (for field size)}}{910}{subsection.59.1.2}
\contentsline {subsection}{\numberline {59.1.3}\leavevmode {\color {Chapter }IsLexOrderedFFE}}{912}{subsection.59.1.3}
\contentsline {section}{\numberline {59.2}\leavevmode {\color {Chapter }Operations for Finite Field Elements}}{912}{section.59.2}
\contentsline {subsection}{\numberline {59.2.1}\leavevmode {\color {Chapter }DegreeFFE (for a FFE)}}{913}{subsection.59.2.1}
\contentsline {subsection}{\numberline {59.2.2}\leavevmode {\color {Chapter }LogFFE}}{913}{subsection.59.2.2}
\contentsline {subsection}{\numberline {59.2.3}\leavevmode {\color {Chapter }IntFFE}}{913}{subsection.59.2.3}
\contentsline {subsection}{\numberline {59.2.4}\leavevmode {\color {Chapter }IntFFESymm (for a FFE)}}{914}{subsection.59.2.4}
\contentsline {subsection}{\numberline {59.2.5}\leavevmode {\color {Chapter }IntVecFFE}}{914}{subsection.59.2.5}
\contentsline {subsection}{\numberline {59.2.6}\leavevmode {\color {Chapter }AsInternalFFE}}{915}{subsection.59.2.6}
\contentsline {section}{\numberline {59.3}\leavevmode {\color {Chapter }Creating Finite Fields}}{915}{section.59.3}
\contentsline {subsection}{\numberline {59.3.1}\leavevmode {\color {Chapter }DefaultField (for finite field elements)}}{915}{subsection.59.3.1}
\contentsline {subsection}{\numberline {59.3.2}\leavevmode {\color {Chapter }GaloisField (for field size)}}{915}{subsection.59.3.2}
\contentsline {subsection}{\numberline {59.3.3}\leavevmode {\color {Chapter }PrimitiveRoot}}{916}{subsection.59.3.3}
\contentsline {section}{\numberline {59.4}\leavevmode {\color {Chapter }Frobenius Automorphisms}}{916}{section.59.4}
\contentsline {subsection}{\numberline {59.4.1}\leavevmode {\color {Chapter }FrobeniusAutomorphism}}{916}{subsection.59.4.1}
\contentsline {section}{\numberline {59.5}\leavevmode {\color {Chapter }Conway Polynomials}}{917}{section.59.5}
\contentsline {subsection}{\numberline {59.5.1}\leavevmode {\color {Chapter }ConwayPolynomial}}{917}{subsection.59.5.1}
\contentsline {subsection}{\numberline {59.5.2}\leavevmode {\color {Chapter }IsCheapConwayPolynomial}}{918}{subsection.59.5.2}
\contentsline {subsection}{\numberline {59.5.3}\leavevmode {\color {Chapter }RandomPrimitivePolynomial}}{918}{subsection.59.5.3}
\contentsline {section}{\numberline {59.6}\leavevmode {\color {Chapter }Printing, Viewing and Displaying Finite Field Elements}}{918}{section.59.6}
\contentsline {subsection}{\numberline {59.6.1}\leavevmode {\color {Chapter }ViewObj (for a ffe)}}{918}{subsection.59.6.1}
\contentsline {chapter}{\numberline {60}\leavevmode {\color {Chapter }Abelian Number Fields}}{920}{chapter.60}
\contentsline {section}{\numberline {60.1}\leavevmode {\color {Chapter }Construction of Abelian Number Fields}}{920}{section.60.1}
\contentsline {subsection}{\numberline {60.1.1}\leavevmode {\color {Chapter }CyclotomicField (for (subfield and) conductor)}}{920}{subsection.60.1.1}
\contentsline {subsection}{\numberline {60.1.2}\leavevmode {\color {Chapter }AbelianNumberField}}{921}{subsection.60.1.2}
\contentsline {subsection}{\numberline {60.1.3}\leavevmode {\color {Chapter }GaussianRationals}}{921}{subsection.60.1.3}
\contentsline {section}{\numberline {60.2}\leavevmode {\color {Chapter }Operations for Abelian Number Fields}}{922}{section.60.2}
\contentsline {subsection}{\numberline {60.2.1}\leavevmode {\color {Chapter }Factors (for polynomials over abelian number fields)}}{922}{subsection.60.2.1}
\contentsline {subsection}{\numberline {60.2.2}\leavevmode {\color {Chapter }IsNumberField}}{922}{subsection.60.2.2}
\contentsline {subsection}{\numberline {60.2.3}\leavevmode {\color {Chapter }IsAbelianNumberField}}{922}{subsection.60.2.3}
\contentsline {subsection}{\numberline {60.2.4}\leavevmode {\color {Chapter }IsCyclotomicField}}{922}{subsection.60.2.4}
\contentsline {subsection}{\numberline {60.2.5}\leavevmode {\color {Chapter }GaloisStabilizer}}{923}{subsection.60.2.5}
\contentsline {section}{\numberline {60.3}\leavevmode {\color {Chapter }Integral Bases of Abelian Number Fields}}{923}{section.60.3}
\contentsline {subsection}{\numberline {60.3.1}\leavevmode {\color {Chapter }ZumbroichBase}}{924}{subsection.60.3.1}
\contentsline {subsection}{\numberline {60.3.2}\leavevmode {\color {Chapter }LenstraBase}}{925}{subsection.60.3.2}
\contentsline {section}{\numberline {60.4}\leavevmode {\color {Chapter }Galois Groups of Abelian Number Fields}}{925}{section.60.4}
\contentsline {subsection}{\numberline {60.4.1}\leavevmode {\color {Chapter }GaloisGroup (for abelian number fields)}}{926}{subsection.60.4.1}
\contentsline {subsection}{\numberline {60.4.2}\leavevmode {\color {Chapter }ANFAutomorphism}}{927}{subsection.60.4.2}
\contentsline {section}{\numberline {60.5}\leavevmode {\color {Chapter }Gaussians}}{927}{section.60.5}
\contentsline {subsection}{\numberline {60.5.1}\leavevmode {\color {Chapter }GaussianIntegers}}{927}{subsection.60.5.1}
\contentsline {subsection}{\numberline {60.5.2}\leavevmode {\color {Chapter }IsGaussianIntegers}}{927}{subsection.60.5.2}
\contentsline {chapter}{\numberline {61}\leavevmode {\color {Chapter }Vector Spaces}}{928}{chapter.61}
\contentsline {section}{\numberline {61.1}\leavevmode {\color {Chapter }IsLeftVectorSpace (Filter)}}{928}{section.61.1}
\contentsline {subsection}{\numberline {61.1.1}\leavevmode {\color {Chapter }IsLeftVectorSpace}}{928}{subsection.61.1.1}
\contentsline {section}{\numberline {61.2}\leavevmode {\color {Chapter }Constructing Vector Spaces}}{928}{section.61.2}
\contentsline {subsection}{\numberline {61.2.1}\leavevmode {\color {Chapter }VectorSpace}}{928}{subsection.61.2.1}
\contentsline {subsection}{\numberline {61.2.2}\leavevmode {\color {Chapter }Subspace}}{929}{subsection.61.2.2}
\contentsline {subsection}{\numberline {61.2.3}\leavevmode {\color {Chapter }AsVectorSpace}}{929}{subsection.61.2.3}
\contentsline {subsection}{\numberline {61.2.4}\leavevmode {\color {Chapter }AsSubspace}}{929}{subsection.61.2.4}
\contentsline {section}{\numberline {61.3}\leavevmode {\color {Chapter }Operations and Attributes for Vector Spaces}}{930}{section.61.3}
\contentsline {subsection}{\numberline {61.3.1}\leavevmode {\color {Chapter }GeneratorsOfLeftVectorSpace}}{930}{subsection.61.3.1}
\contentsline {subsection}{\numberline {61.3.2}\leavevmode {\color {Chapter }TrivialSubspace}}{930}{subsection.61.3.2}
\contentsline {section}{\numberline {61.4}\leavevmode {\color {Chapter }Domains of Subspaces of Vector Spaces}}{930}{section.61.4}
\contentsline {subsection}{\numberline {61.4.1}\leavevmode {\color {Chapter }Subspaces}}{930}{subsection.61.4.1}
\contentsline {subsection}{\numberline {61.4.2}\leavevmode {\color {Chapter }IsSubspacesVectorSpace}}{931}{subsection.61.4.2}
\contentsline {section}{\numberline {61.5}\leavevmode {\color {Chapter }Bases of Vector Spaces}}{931}{section.61.5}
\contentsline {subsection}{\numberline {61.5.1}\leavevmode {\color {Chapter }IsBasis}}{932}{subsection.61.5.1}
\contentsline {subsection}{\numberline {61.5.2}\leavevmode {\color {Chapter }Basis}}{932}{subsection.61.5.2}
\contentsline {subsection}{\numberline {61.5.3}\leavevmode {\color {Chapter }CanonicalBasis}}{933}{subsection.61.5.3}
\contentsline {subsection}{\numberline {61.5.4}\leavevmode {\color {Chapter }RelativeBasis}}{933}{subsection.61.5.4}
\contentsline {section}{\numberline {61.6}\leavevmode {\color {Chapter }Operations for Vector Space Bases}}{933}{section.61.6}
\contentsline {subsection}{\numberline {61.6.1}\leavevmode {\color {Chapter }BasisVectors}}{933}{subsection.61.6.1}
\contentsline {subsection}{\numberline {61.6.2}\leavevmode {\color {Chapter }UnderlyingLeftModule}}{934}{subsection.61.6.2}
\contentsline {subsection}{\numberline {61.6.3}\leavevmode {\color {Chapter }Coefficients}}{934}{subsection.61.6.3}
\contentsline {subsection}{\numberline {61.6.4}\leavevmode {\color {Chapter }LinearCombination}}{934}{subsection.61.6.4}
\contentsline {subsection}{\numberline {61.6.5}\leavevmode {\color {Chapter }EnumeratorByBasis}}{935}{subsection.61.6.5}
\contentsline {subsection}{\numberline {61.6.6}\leavevmode {\color {Chapter }IteratorByBasis}}{935}{subsection.61.6.6}
\contentsline {section}{\numberline {61.7}\leavevmode {\color {Chapter }Operations for Special Kinds of Bases}}{935}{section.61.7}
\contentsline {subsection}{\numberline {61.7.1}\leavevmode {\color {Chapter }IsCanonicalBasis}}{935}{subsection.61.7.1}
\contentsline {subsection}{\numberline {61.7.2}\leavevmode {\color {Chapter }IsIntegralBasis}}{936}{subsection.61.7.2}
\contentsline {subsection}{\numberline {61.7.3}\leavevmode {\color {Chapter }IsNormalBasis}}{936}{subsection.61.7.3}
\contentsline {section}{\numberline {61.8}\leavevmode {\color {Chapter }Mutable Bases}}{936}{section.61.8}
\contentsline {subsection}{\numberline {61.8.1}\leavevmode {\color {Chapter }IsMutableBasis}}{937}{subsection.61.8.1}
\contentsline {subsection}{\numberline {61.8.2}\leavevmode {\color {Chapter }MutableBasis}}{937}{subsection.61.8.2}
\contentsline {subsection}{\numberline {61.8.3}\leavevmode {\color {Chapter }NrBasisVectors}}{937}{subsection.61.8.3}
\contentsline {subsection}{\numberline {61.8.4}\leavevmode {\color {Chapter }ImmutableBasis}}{937}{subsection.61.8.4}
\contentsline {subsection}{\numberline {61.8.5}\leavevmode {\color {Chapter }IsContainedInSpan}}{938}{subsection.61.8.5}
\contentsline {subsection}{\numberline {61.8.6}\leavevmode {\color {Chapter }CloseMutableBasis}}{938}{subsection.61.8.6}
\contentsline {section}{\numberline {61.9}\leavevmode {\color {Chapter }Row and Matrix Spaces}}{939}{section.61.9}
\contentsline {subsection}{\numberline {61.9.1}\leavevmode {\color {Chapter }IsRowSpace}}{939}{subsection.61.9.1}
\contentsline {subsection}{\numberline {61.9.2}\leavevmode {\color {Chapter }IsMatrixSpace}}{939}{subsection.61.9.2}
\contentsline {subsection}{\numberline {61.9.3}\leavevmode {\color {Chapter }IsGaussianSpace}}{939}{subsection.61.9.3}
\contentsline {subsection}{\numberline {61.9.4}\leavevmode {\color {Chapter }FullRowSpace}}{939}{subsection.61.9.4}
\contentsline {subsection}{\numberline {61.9.5}\leavevmode {\color {Chapter }FullMatrixSpace}}{940}{subsection.61.9.5}
\contentsline {subsection}{\numberline {61.9.6}\leavevmode {\color {Chapter }DimensionOfVectors}}{940}{subsection.61.9.6}
\contentsline {subsection}{\numberline {61.9.7}\leavevmode {\color {Chapter }IsSemiEchelonized}}{940}{subsection.61.9.7}
\contentsline {subsection}{\numberline {61.9.8}\leavevmode {\color {Chapter }SemiEchelonBasis}}{941}{subsection.61.9.8}
\contentsline {subsection}{\numberline {61.9.9}\leavevmode {\color {Chapter }IsCanonicalBasisFullRowModule}}{941}{subsection.61.9.9}
\contentsline {subsection}{\numberline {61.9.10}\leavevmode {\color {Chapter }IsCanonicalBasisFullMatrixModule}}{941}{subsection.61.9.10}
\contentsline {subsection}{\numberline {61.9.11}\leavevmode {\color {Chapter }NormedRowVectors}}{942}{subsection.61.9.11}
\contentsline {subsection}{\numberline {61.9.12}\leavevmode {\color {Chapter }SiftedVector}}{942}{subsection.61.9.12}
\contentsline {section}{\numberline {61.10}\leavevmode {\color {Chapter }Vector Space Homomorphisms}}{942}{section.61.10}
\contentsline {subsection}{\numberline {61.10.1}\leavevmode {\color {Chapter }LeftModuleGeneralMappingByImages}}{942}{subsection.61.10.1}
\contentsline {subsection}{\numberline {61.10.2}\leavevmode {\color {Chapter }LeftModuleHomomorphismByImages}}{943}{subsection.61.10.2}
\contentsline {subsection}{\numberline {61.10.3}\leavevmode {\color {Chapter }LeftModuleHomomorphismByMatrix}}{943}{subsection.61.10.3}
\contentsline {subsection}{\numberline {61.10.4}\leavevmode {\color {Chapter }NaturalHomomorphismBySubspace}}{944}{subsection.61.10.4}
\contentsline {subsection}{\numberline {61.10.5}\leavevmode {\color {Chapter }Hom}}{944}{subsection.61.10.5}
\contentsline {subsection}{\numberline {61.10.6}\leavevmode {\color {Chapter }End}}{944}{subsection.61.10.6}
\contentsline {subsection}{\numberline {61.10.7}\leavevmode {\color {Chapter }IsFullHomModule}}{945}{subsection.61.10.7}
\contentsline {subsection}{\numberline {61.10.8}\leavevmode {\color {Chapter }IsPseudoCanonicalBasisFullHomModule}}{945}{subsection.61.10.8}
\contentsline {subsection}{\numberline {61.10.9}\leavevmode {\color {Chapter }IsLinearMappingsModule}}{945}{subsection.61.10.9}
\contentsline {section}{\numberline {61.11}\leavevmode {\color {Chapter }Vector Spaces Handled By Nice Bases}}{945}{section.61.11}
\contentsline {subsection}{\numberline {61.11.1}\leavevmode {\color {Chapter }NiceFreeLeftModule}}{946}{subsection.61.11.1}
\contentsline {subsection}{\numberline {61.11.2}\leavevmode {\color {Chapter }NiceVector}}{946}{subsection.61.11.2}
\contentsline {subsection}{\numberline {61.11.3}\leavevmode {\color {Chapter }NiceFreeLeftModuleInfo}}{946}{subsection.61.11.3}
\contentsline {subsection}{\numberline {61.11.4}\leavevmode {\color {Chapter }NiceBasis}}{946}{subsection.61.11.4}
\contentsline {subsection}{\numberline {61.11.5}\leavevmode {\color {Chapter }IsBasisByNiceBasis}}{947}{subsection.61.11.5}
\contentsline {subsection}{\numberline {61.11.6}\leavevmode {\color {Chapter }IsHandledByNiceBasis}}{947}{subsection.61.11.6}
\contentsline {section}{\numberline {61.12}\leavevmode {\color {Chapter }How to Implement New Kinds of Vector Spaces}}{947}{section.61.12}
\contentsline {subsection}{\numberline {61.12.1}\leavevmode {\color {Chapter }DeclareHandlingByNiceBasis}}{947}{subsection.61.12.1}
\contentsline {subsection}{\numberline {61.12.2}\leavevmode {\color {Chapter }NiceBasisFiltersInfo}}{948}{subsection.61.12.2}
\contentsline {subsection}{\numberline {61.12.3}\leavevmode {\color {Chapter }CheckForHandlingByNiceBasis}}{948}{subsection.61.12.3}
\contentsline {chapter}{\numberline {62}\leavevmode {\color {Chapter }Algebras}}{949}{chapter.62}
\contentsline {section}{\numberline {62.1}\leavevmode {\color {Chapter }InfoAlgebra (Info Class)}}{949}{section.62.1}
\contentsline {subsection}{\numberline {62.1.1}\leavevmode {\color {Chapter }InfoAlgebra}}{949}{subsection.62.1.1}
\contentsline {section}{\numberline {62.2}\leavevmode {\color {Chapter }Constructing Algebras by Generators}}{949}{section.62.2}
\contentsline {subsection}{\numberline {62.2.1}\leavevmode {\color {Chapter }Algebra}}{949}{subsection.62.2.1}
\contentsline {subsection}{\numberline {62.2.2}\leavevmode {\color {Chapter }AlgebraWithOne}}{950}{subsection.62.2.2}
\contentsline {section}{\numberline {62.3}\leavevmode {\color {Chapter }Constructing Algebras as Free Algebras}}{950}{section.62.3}
\contentsline {subsection}{\numberline {62.3.1}\leavevmode {\color {Chapter }FreeAlgebra (for ring, rank (and name))}}{950}{subsection.62.3.1}
\contentsline {subsection}{\numberline {62.3.2}\leavevmode {\color {Chapter }FreeAlgebraWithOne (for ring, rank (and name))}}{950}{subsection.62.3.2}
\contentsline {subsection}{\numberline {62.3.3}\leavevmode {\color {Chapter }FreeAssociativeAlgebra (for ring, rank (and name))}}{951}{subsection.62.3.3}
\contentsline {subsection}{\numberline {62.3.4}\leavevmode {\color {Chapter }FreeAssociativeAlgebraWithOne (for ring, rank (and name))}}{951}{subsection.62.3.4}
\contentsline {section}{\numberline {62.4}\leavevmode {\color {Chapter }Constructing Algebras by Structure Constants}}{951}{section.62.4}
\contentsline {subsection}{\numberline {62.4.1}\leavevmode {\color {Chapter }AlgebraByStructureConstants}}{951}{subsection.62.4.1}
\contentsline {subsection}{\numberline {62.4.2}\leavevmode {\color {Chapter }StructureConstantsTable}}{952}{subsection.62.4.2}
\contentsline {subsection}{\numberline {62.4.3}\leavevmode {\color {Chapter }EmptySCTable}}{952}{subsection.62.4.3}
\contentsline {subsection}{\numberline {62.4.4}\leavevmode {\color {Chapter }SetEntrySCTable}}{953}{subsection.62.4.4}
\contentsline {subsection}{\numberline {62.4.5}\leavevmode {\color {Chapter }GapInputSCTable}}{953}{subsection.62.4.5}
\contentsline {subsection}{\numberline {62.4.6}\leavevmode {\color {Chapter }TestJacobi}}{953}{subsection.62.4.6}
\contentsline {subsection}{\numberline {62.4.7}\leavevmode {\color {Chapter }IdentityFromSCTable}}{954}{subsection.62.4.7}
\contentsline {subsection}{\numberline {62.4.8}\leavevmode {\color {Chapter }QuotientFromSCTable}}{954}{subsection.62.4.8}
\contentsline {section}{\numberline {62.5}\leavevmode {\color {Chapter }Some Special Algebras}}{954}{section.62.5}
\contentsline {subsection}{\numberline {62.5.1}\leavevmode {\color {Chapter }QuaternionAlgebra}}{954}{subsection.62.5.1}
\contentsline {subsection}{\numberline {62.5.2}\leavevmode {\color {Chapter }ComplexificationQuat (for a vector)}}{955}{subsection.62.5.2}
\contentsline {subsection}{\numberline {62.5.3}\leavevmode {\color {Chapter }OctaveAlgebra}}{955}{subsection.62.5.3}
\contentsline {subsection}{\numberline {62.5.4}\leavevmode {\color {Chapter }FullMatrixAlgebra}}{955}{subsection.62.5.4}
\contentsline {subsection}{\numberline {62.5.5}\leavevmode {\color {Chapter }NullAlgebra}}{956}{subsection.62.5.5}
\contentsline {section}{\numberline {62.6}\leavevmode {\color {Chapter }Subalgebras}}{956}{section.62.6}
\contentsline {subsection}{\numberline {62.6.1}\leavevmode {\color {Chapter }Subalgebra}}{956}{subsection.62.6.1}
\contentsline {subsection}{\numberline {62.6.2}\leavevmode {\color {Chapter }SubalgebraNC}}{956}{subsection.62.6.2}
\contentsline {subsection}{\numberline {62.6.3}\leavevmode {\color {Chapter }SubalgebraWithOne}}{957}{subsection.62.6.3}
\contentsline {subsection}{\numberline {62.6.4}\leavevmode {\color {Chapter }SubalgebraWithOneNC}}{957}{subsection.62.6.4}
\contentsline {subsection}{\numberline {62.6.5}\leavevmode {\color {Chapter }TrivialSubalgebra}}{957}{subsection.62.6.5}
\contentsline {section}{\numberline {62.7}\leavevmode {\color {Chapter }Ideals of Algebras}}{958}{section.62.7}
\contentsline {section}{\numberline {62.8}\leavevmode {\color {Chapter }Categories and Properties of Algebras}}{958}{section.62.8}
\contentsline {subsection}{\numberline {62.8.1}\leavevmode {\color {Chapter }IsFLMLOR}}{958}{subsection.62.8.1}
\contentsline {subsection}{\numberline {62.8.2}\leavevmode {\color {Chapter }IsFLMLORWithOne}}{959}{subsection.62.8.2}
\contentsline {subsection}{\numberline {62.8.3}\leavevmode {\color {Chapter }IsAlgebra}}{959}{subsection.62.8.3}
\contentsline {subsection}{\numberline {62.8.4}\leavevmode {\color {Chapter }IsAlgebraWithOne}}{959}{subsection.62.8.4}
\contentsline {subsection}{\numberline {62.8.5}\leavevmode {\color {Chapter }IsLieAlgebra}}{959}{subsection.62.8.5}
\contentsline {subsection}{\numberline {62.8.6}\leavevmode {\color {Chapter }IsSimpleAlgebra}}{960}{subsection.62.8.6}
\contentsline {subsection}{\numberline {62.8.7}\leavevmode {\color {Chapter }IsFiniteDimensional (for matrix algebras)}}{960}{subsection.62.8.7}
\contentsline {subsection}{\numberline {62.8.8}\leavevmode {\color {Chapter }IsQuaternion}}{960}{subsection.62.8.8}
\contentsline {section}{\numberline {62.9}\leavevmode {\color {Chapter }Attributes and Operations for Algebras}}{961}{section.62.9}
\contentsline {subsection}{\numberline {62.9.1}\leavevmode {\color {Chapter }GeneratorsOfAlgebra}}{961}{subsection.62.9.1}
\contentsline {subsection}{\numberline {62.9.2}\leavevmode {\color {Chapter }GeneratorsOfAlgebraWithOne}}{961}{subsection.62.9.2}
\contentsline {subsection}{\numberline {62.9.3}\leavevmode {\color {Chapter }ProductSpace}}{961}{subsection.62.9.3}
\contentsline {subsection}{\numberline {62.9.4}\leavevmode {\color {Chapter }PowerSubalgebraSeries}}{962}{subsection.62.9.4}
\contentsline {subsection}{\numberline {62.9.5}\leavevmode {\color {Chapter }AdjointBasis}}{962}{subsection.62.9.5}
\contentsline {subsection}{\numberline {62.9.6}\leavevmode {\color {Chapter }IndicesOfAdjointBasis}}{962}{subsection.62.9.6}
\contentsline {subsection}{\numberline {62.9.7}\leavevmode {\color {Chapter }AsAlgebra}}{962}{subsection.62.9.7}
\contentsline {subsection}{\numberline {62.9.8}\leavevmode {\color {Chapter }AsAlgebraWithOne}}{963}{subsection.62.9.8}
\contentsline {subsection}{\numberline {62.9.9}\leavevmode {\color {Chapter }AsSubalgebra}}{963}{subsection.62.9.9}
\contentsline {subsection}{\numberline {62.9.10}\leavevmode {\color {Chapter }AsSubalgebraWithOne}}{963}{subsection.62.9.10}
\contentsline {subsection}{\numberline {62.9.11}\leavevmode {\color {Chapter }MutableBasisOfClosureUnderAction}}{963}{subsection.62.9.11}
\contentsline {subsection}{\numberline {62.9.12}\leavevmode {\color {Chapter }MutableBasisOfNonassociativeAlgebra}}{964}{subsection.62.9.12}
\contentsline {subsection}{\numberline {62.9.13}\leavevmode {\color {Chapter }MutableBasisOfIdealInNonassociativeAlgebra}}{965}{subsection.62.9.13}
\contentsline {subsection}{\numberline {62.9.14}\leavevmode {\color {Chapter }DirectSumOfAlgebras (for two algebras)}}{965}{subsection.62.9.14}
\contentsline {subsection}{\numberline {62.9.15}\leavevmode {\color {Chapter }FullMatrixAlgebraCentralizer}}{965}{subsection.62.9.15}
\contentsline {subsection}{\numberline {62.9.16}\leavevmode {\color {Chapter }RadicalOfAlgebra}}{966}{subsection.62.9.16}
\contentsline {subsection}{\numberline {62.9.17}\leavevmode {\color {Chapter }CentralIdempotentsOfAlgebra}}{966}{subsection.62.9.17}
\contentsline {subsection}{\numberline {62.9.18}\leavevmode {\color {Chapter }DirectSumDecomposition (for Lie algebras)}}{966}{subsection.62.9.18}
\contentsline {subsection}{\numberline {62.9.19}\leavevmode {\color {Chapter }LeviMalcevDecomposition (for Lie algebras)}}{967}{subsection.62.9.19}
\contentsline {subsection}{\numberline {62.9.20}\leavevmode {\color {Chapter }Grading}}{967}{subsection.62.9.20}
\contentsline {section}{\numberline {62.10}\leavevmode {\color {Chapter }Homomorphisms of Algebras}}{968}{section.62.10}
\contentsline {subsection}{\numberline {62.10.1}\leavevmode {\color {Chapter }AlgebraGeneralMappingByImages}}{969}{subsection.62.10.1}
\contentsline {subsection}{\numberline {62.10.2}\leavevmode {\color {Chapter }AlgebraHomomorphismByImages}}{969}{subsection.62.10.2}
\contentsline {subsection}{\numberline {62.10.3}\leavevmode {\color {Chapter }AlgebraHomomorphismByImagesNC}}{969}{subsection.62.10.3}
\contentsline {subsection}{\numberline {62.10.4}\leavevmode {\color {Chapter }AlgebraWithOneGeneralMappingByImages}}{970}{subsection.62.10.4}
\contentsline {subsection}{\numberline {62.10.5}\leavevmode {\color {Chapter }AlgebraWithOneHomomorphismByImages}}{970}{subsection.62.10.5}
\contentsline {subsection}{\numberline {62.10.6}\leavevmode {\color {Chapter }AlgebraWithOneHomomorphismByImagesNC}}{971}{subsection.62.10.6}
\contentsline {subsection}{\numberline {62.10.7}\leavevmode {\color {Chapter }NaturalHomomorphismByIdeal (for an algebra and an ideal)}}{971}{subsection.62.10.7}
\contentsline {subsection}{\numberline {62.10.8}\leavevmode {\color {Chapter }OperationAlgebraHomomorphism (action w.r.t. a basis of the module)}}{971}{subsection.62.10.8}
\contentsline {subsection}{\numberline {62.10.9}\leavevmode {\color {Chapter }NiceAlgebraMonomorphism}}{972}{subsection.62.10.9}
\contentsline {subsection}{\numberline {62.10.10}\leavevmode {\color {Chapter }IsomorphismFpAlgebra}}{972}{subsection.62.10.10}
\contentsline {subsection}{\numberline {62.10.11}\leavevmode {\color {Chapter }IsomorphismMatrixAlgebra}}{972}{subsection.62.10.11}
\contentsline {subsection}{\numberline {62.10.12}\leavevmode {\color {Chapter }IsomorphismSCAlgebra (w.r.t. a given basis)}}{973}{subsection.62.10.12}
\contentsline {subsection}{\numberline {62.10.13}\leavevmode {\color {Chapter }RepresentativeLinearOperation}}{973}{subsection.62.10.13}
\contentsline {section}{\numberline {62.11}\leavevmode {\color {Chapter }Representations of Algebras}}{973}{section.62.11}
\contentsline {subsection}{\numberline {62.11.1}\leavevmode {\color {Chapter }LeftAlgebraModuleByGenerators}}{974}{subsection.62.11.1}
\contentsline {subsection}{\numberline {62.11.2}\leavevmode {\color {Chapter }RightAlgebraModuleByGenerators}}{974}{subsection.62.11.2}
\contentsline {subsection}{\numberline {62.11.3}\leavevmode {\color {Chapter }BiAlgebraModuleByGenerators}}{974}{subsection.62.11.3}
\contentsline {subsection}{\numberline {62.11.4}\leavevmode {\color {Chapter }LeftAlgebraModule}}{975}{subsection.62.11.4}
\contentsline {subsection}{\numberline {62.11.5}\leavevmode {\color {Chapter }RightAlgebraModule}}{975}{subsection.62.11.5}
\contentsline {subsection}{\numberline {62.11.6}\leavevmode {\color {Chapter }BiAlgebraModule}}{975}{subsection.62.11.6}
\contentsline {subsection}{\numberline {62.11.7}\leavevmode {\color {Chapter }GeneratorsOfAlgebraModule}}{976}{subsection.62.11.7}
\contentsline {subsection}{\numberline {62.11.8}\leavevmode {\color {Chapter }IsAlgebraModuleElement}}{976}{subsection.62.11.8}
\contentsline {subsection}{\numberline {62.11.9}\leavevmode {\color {Chapter }IsLeftAlgebraModuleElement}}{976}{subsection.62.11.9}
\contentsline {subsection}{\numberline {62.11.10}\leavevmode {\color {Chapter }IsRightAlgebraModuleElement}}{976}{subsection.62.11.10}
\contentsline {subsection}{\numberline {62.11.11}\leavevmode {\color {Chapter }LeftActingAlgebra}}{977}{subsection.62.11.11}
\contentsline {subsection}{\numberline {62.11.12}\leavevmode {\color {Chapter }RightActingAlgebra}}{977}{subsection.62.11.12}
\contentsline {subsection}{\numberline {62.11.13}\leavevmode {\color {Chapter }ActingAlgebra}}{977}{subsection.62.11.13}
\contentsline {subsection}{\numberline {62.11.14}\leavevmode {\color {Chapter }IsBasisOfAlgebraModuleElementSpace}}{977}{subsection.62.11.14}
\contentsline {subsection}{\numberline {62.11.15}\leavevmode {\color {Chapter }MatrixOfAction}}{978}{subsection.62.11.15}
\contentsline {subsection}{\numberline {62.11.16}\leavevmode {\color {Chapter }SubAlgebraModule}}{978}{subsection.62.11.16}
\contentsline {subsection}{\numberline {62.11.17}\leavevmode {\color {Chapter }LeftModuleByHomomorphismToMatAlg}}{979}{subsection.62.11.17}
\contentsline {subsection}{\numberline {62.11.18}\leavevmode {\color {Chapter }RightModuleByHomomorphismToMatAlg}}{979}{subsection.62.11.18}
\contentsline {subsection}{\numberline {62.11.19}\leavevmode {\color {Chapter }AdjointModule}}{979}{subsection.62.11.19}
\contentsline {subsection}{\numberline {62.11.20}\leavevmode {\color {Chapter }FaithfulModule (for Lie algebras)}}{980}{subsection.62.11.20}
\contentsline {subsection}{\numberline {62.11.21}\leavevmode {\color {Chapter }ModuleByRestriction}}{980}{subsection.62.11.21}
\contentsline {subsection}{\numberline {62.11.22}\leavevmode {\color {Chapter }NaturalHomomorphismBySubAlgebraModule}}{981}{subsection.62.11.22}
\contentsline {subsection}{\numberline {62.11.23}\leavevmode {\color {Chapter }DirectSumOfAlgebraModules (for a list of Lie algebra modules)}}{982}{subsection.62.11.23}
\contentsline {subsection}{\numberline {62.11.24}\leavevmode {\color {Chapter }TranslatorSubalgebra}}{982}{subsection.62.11.24}
\contentsline {chapter}{\numberline {63}\leavevmode {\color {Chapter }Finitely Presented Algebras}}{984}{chapter.63}
\contentsline {chapter}{\numberline {64}\leavevmode {\color {Chapter }Lie Algebras}}{985}{chapter.64}
\contentsline {section}{\numberline {64.1}\leavevmode {\color {Chapter }Lie Objects}}{985}{section.64.1}
\contentsline {subsection}{\numberline {64.1.1}\leavevmode {\color {Chapter }LieObject}}{986}{subsection.64.1.1}
\contentsline {subsection}{\numberline {64.1.2}\leavevmode {\color {Chapter }IsLieObject}}{986}{subsection.64.1.2}
\contentsline {subsection}{\numberline {64.1.3}\leavevmode {\color {Chapter }LieFamily}}{986}{subsection.64.1.3}
\contentsline {subsection}{\numberline {64.1.4}\leavevmode {\color {Chapter }UnderlyingFamily}}{987}{subsection.64.1.4}
\contentsline {subsection}{\numberline {64.1.5}\leavevmode {\color {Chapter }UnderlyingRingElement}}{987}{subsection.64.1.5}
\contentsline {section}{\numberline {64.2}\leavevmode {\color {Chapter }Constructing Lie algebras}}{987}{section.64.2}
\contentsline {subsection}{\numberline {64.2.1}\leavevmode {\color {Chapter }LieAlgebraByStructureConstants}}{987}{subsection.64.2.1}
\contentsline {subsection}{\numberline {64.2.2}\leavevmode {\color {Chapter }RestrictedLieAlgebraByStructureConstants}}{988}{subsection.64.2.2}
\contentsline {subsection}{\numberline {64.2.3}\leavevmode {\color {Chapter }LieAlgebra (for an associative algebra)}}{988}{subsection.64.2.3}
\contentsline {subsection}{\numberline {64.2.4}\leavevmode {\color {Chapter }FreeLieAlgebra (for ring, rank (and name))}}{989}{subsection.64.2.4}
\contentsline {subsection}{\numberline {64.2.5}\leavevmode {\color {Chapter }FullMatrixLieAlgebra}}{989}{subsection.64.2.5}
\contentsline {subsection}{\numberline {64.2.6}\leavevmode {\color {Chapter }RightDerivations}}{989}{subsection.64.2.6}
\contentsline {subsection}{\numberline {64.2.7}\leavevmode {\color {Chapter }SimpleLieAlgebra}}{990}{subsection.64.2.7}
\contentsline {section}{\numberline {64.3}\leavevmode {\color {Chapter }Distinguished Subalgebras}}{990}{section.64.3}
\contentsline {subsection}{\numberline {64.3.1}\leavevmode {\color {Chapter }LieCentre}}{991}{subsection.64.3.1}
\contentsline {subsection}{\numberline {64.3.2}\leavevmode {\color {Chapter }LieCentralizer}}{991}{subsection.64.3.2}
\contentsline {subsection}{\numberline {64.3.3}\leavevmode {\color {Chapter }LieNormalizer}}{991}{subsection.64.3.3}
\contentsline {subsection}{\numberline {64.3.4}\leavevmode {\color {Chapter }LieDerivedSubalgebra}}{991}{subsection.64.3.4}
\contentsline {subsection}{\numberline {64.3.5}\leavevmode {\color {Chapter }LieNilRadical}}{992}{subsection.64.3.5}
\contentsline {subsection}{\numberline {64.3.6}\leavevmode {\color {Chapter }LieSolvableRadical}}{992}{subsection.64.3.6}
\contentsline {subsection}{\numberline {64.3.7}\leavevmode {\color {Chapter }CartanSubalgebra}}{992}{subsection.64.3.7}
\contentsline {section}{\numberline {64.4}\leavevmode {\color {Chapter }Series of Ideals}}{992}{section.64.4}
\contentsline {subsection}{\numberline {64.4.1}\leavevmode {\color {Chapter }LieDerivedSeries}}{992}{subsection.64.4.1}
\contentsline {subsection}{\numberline {64.4.2}\leavevmode {\color {Chapter }LieLowerCentralSeries}}{993}{subsection.64.4.2}
\contentsline {subsection}{\numberline {64.4.3}\leavevmode {\color {Chapter }LieUpperCentralSeries}}{993}{subsection.64.4.3}
\contentsline {section}{\numberline {64.5}\leavevmode {\color {Chapter }Properties of a Lie Algebra}}{993}{section.64.5}
\contentsline {subsection}{\numberline {64.5.1}\leavevmode {\color {Chapter }IsLieAbelian}}{993}{subsection.64.5.1}
\contentsline {subsection}{\numberline {64.5.2}\leavevmode {\color {Chapter }IsLieNilpotent}}{994}{subsection.64.5.2}
\contentsline {subsection}{\numberline {64.5.3}\leavevmode {\color {Chapter }IsLieSolvable}}{994}{subsection.64.5.3}
\contentsline {section}{\numberline {64.6}\leavevmode {\color {Chapter }Semisimple Lie Algebras and Root Systems}}{994}{section.64.6}
\contentsline {subsection}{\numberline {64.6.1}\leavevmode {\color {Chapter }SemiSimpleType}}{994}{subsection.64.6.1}
\contentsline {subsection}{\numberline {64.6.2}\leavevmode {\color {Chapter }ChevalleyBasis}}{995}{subsection.64.6.2}
\contentsline {subsection}{\numberline {64.6.3}\leavevmode {\color {Chapter }IsRootSystem}}{995}{subsection.64.6.3}
\contentsline {subsection}{\numberline {64.6.4}\leavevmode {\color {Chapter }IsRootSystemFromLieAlgebra}}{995}{subsection.64.6.4}
\contentsline {subsection}{\numberline {64.6.5}\leavevmode {\color {Chapter }RootSystem}}{995}{subsection.64.6.5}
\contentsline {subsection}{\numberline {64.6.6}\leavevmode {\color {Chapter }UnderlyingLieAlgebra}}{995}{subsection.64.6.6}
\contentsline {subsection}{\numberline {64.6.7}\leavevmode {\color {Chapter }PositiveRoots}}{996}{subsection.64.6.7}
\contentsline {subsection}{\numberline {64.6.8}\leavevmode {\color {Chapter }NegativeRoots}}{996}{subsection.64.6.8}
\contentsline {subsection}{\numberline {64.6.9}\leavevmode {\color {Chapter }PositiveRootVectors}}{996}{subsection.64.6.9}
\contentsline {subsection}{\numberline {64.6.10}\leavevmode {\color {Chapter }NegativeRootVectors}}{996}{subsection.64.6.10}
\contentsline {subsection}{\numberline {64.6.11}\leavevmode {\color {Chapter }SimpleSystem}}{996}{subsection.64.6.11}
\contentsline {subsection}{\numberline {64.6.12}\leavevmode {\color {Chapter }CartanMatrix}}{996}{subsection.64.6.12}
\contentsline {subsection}{\numberline {64.6.13}\leavevmode {\color {Chapter }BilinearFormMat}}{996}{subsection.64.6.13}
\contentsline {subsection}{\numberline {64.6.14}\leavevmode {\color {Chapter }CanonicalGenerators}}{997}{subsection.64.6.14}
\contentsline {section}{\numberline {64.7}\leavevmode {\color {Chapter }Semisimple Lie Algebras and Weyl Groups of Root Systems}}{997}{section.64.7}
\contentsline {subsection}{\numberline {64.7.1}\leavevmode {\color {Chapter }IsWeylGroup}}{997}{subsection.64.7.1}
\contentsline {subsection}{\numberline {64.7.2}\leavevmode {\color {Chapter }SparseCartanMatrix}}{998}{subsection.64.7.2}
\contentsline {subsection}{\numberline {64.7.3}\leavevmode {\color {Chapter }WeylGroup}}{998}{subsection.64.7.3}
\contentsline {subsection}{\numberline {64.7.4}\leavevmode {\color {Chapter }ApplySimpleReflection}}{998}{subsection.64.7.4}
\contentsline {subsection}{\numberline {64.7.5}\leavevmode {\color {Chapter }LongestWeylWordPerm}}{999}{subsection.64.7.5}
\contentsline {subsection}{\numberline {64.7.6}\leavevmode {\color {Chapter }ConjugateDominantWeight}}{999}{subsection.64.7.6}
\contentsline {subsection}{\numberline {64.7.7}\leavevmode {\color {Chapter }WeylOrbitIterator}}{999}{subsection.64.7.7}
\contentsline {section}{\numberline {64.8}\leavevmode {\color {Chapter }Restricted Lie algebras}}{1000}{section.64.8}
\contentsline {subsection}{\numberline {64.8.1}\leavevmode {\color {Chapter }IsRestrictedLieAlgebra}}{1000}{subsection.64.8.1}
\contentsline {subsection}{\numberline {64.8.2}\leavevmode {\color {Chapter }PthPowerImages}}{1000}{subsection.64.8.2}
\contentsline {subsection}{\numberline {64.8.3}\leavevmode {\color {Chapter }PthPowerImage (for basis and element)}}{1001}{subsection.64.8.3}
\contentsline {subsection}{\numberline {64.8.4}\leavevmode {\color {Chapter }JenningsLieAlgebra}}{1001}{subsection.64.8.4}
\contentsline {subsection}{\numberline {64.8.5}\leavevmode {\color {Chapter }PCentralLieAlgebra}}{1001}{subsection.64.8.5}
\contentsline {subsection}{\numberline {64.8.6}\leavevmode {\color {Chapter }NaturalHomomorphismOfLieAlgebraFromNilpotentGroup}}{1002}{subsection.64.8.6}
\contentsline {section}{\numberline {64.9}\leavevmode {\color {Chapter }The Adjoint Representation}}{1002}{section.64.9}
\contentsline {subsection}{\numberline {64.9.1}\leavevmode {\color {Chapter }AdjointMatrix}}{1002}{subsection.64.9.1}
\contentsline {subsection}{\numberline {64.9.2}\leavevmode {\color {Chapter }AdjointAssociativeAlgebra}}{1003}{subsection.64.9.2}
\contentsline {subsection}{\numberline {64.9.3}\leavevmode {\color {Chapter }KillingMatrix}}{1003}{subsection.64.9.3}
\contentsline {subsection}{\numberline {64.9.4}\leavevmode {\color {Chapter }KappaPerp}}{1003}{subsection.64.9.4}
\contentsline {subsection}{\numberline {64.9.5}\leavevmode {\color {Chapter }IsNilpotentElement}}{1003}{subsection.64.9.5}
\contentsline {subsection}{\numberline {64.9.6}\leavevmode {\color {Chapter }NonNilpotentElement}}{1004}{subsection.64.9.6}
\contentsline {subsection}{\numberline {64.9.7}\leavevmode {\color {Chapter }FindSl2}}{1004}{subsection.64.9.7}
\contentsline {section}{\numberline {64.10}\leavevmode {\color {Chapter }Universal Enveloping Algebras}}{1004}{section.64.10}
\contentsline {subsection}{\numberline {64.10.1}\leavevmode {\color {Chapter }UniversalEnvelopingAlgebra}}{1004}{subsection.64.10.1}
\contentsline {section}{\numberline {64.11}\leavevmode {\color {Chapter }Finitely Presented Lie Algebras}}{1005}{section.64.11}
\contentsline {subsection}{\numberline {64.11.1}\leavevmode {\color {Chapter }FpLieAlgebraByCartanMatrix}}{1005}{subsection.64.11.1}
\contentsline {subsection}{\numberline {64.11.2}\leavevmode {\color {Chapter }NilpotentQuotientOfFpLieAlgebra}}{1005}{subsection.64.11.2}
\contentsline {section}{\numberline {64.12}\leavevmode {\color {Chapter }Modules over Lie Algebras and Their Cohomology}}{1006}{section.64.12}
\contentsline {subsection}{\numberline {64.12.1}\leavevmode {\color {Chapter }IsCochain}}{1007}{subsection.64.12.1}
\contentsline {subsection}{\numberline {64.12.2}\leavevmode {\color {Chapter }Cochain}}{1007}{subsection.64.12.2}
\contentsline {subsection}{\numberline {64.12.3}\leavevmode {\color {Chapter }CochainSpace}}{1007}{subsection.64.12.3}
\contentsline {subsection}{\numberline {64.12.4}\leavevmode {\color {Chapter }ValueCochain}}{1008}{subsection.64.12.4}
\contentsline {subsection}{\numberline {64.12.5}\leavevmode {\color {Chapter }LieCoboundaryOperator}}{1008}{subsection.64.12.5}
\contentsline {subsection}{\numberline {64.12.6}\leavevmode {\color {Chapter }Cocycles (for Lie algebra module)}}{1008}{subsection.64.12.6}
\contentsline {subsection}{\numberline {64.12.7}\leavevmode {\color {Chapter }Coboundaries}}{1008}{subsection.64.12.7}
\contentsline {section}{\numberline {64.13}\leavevmode {\color {Chapter }Modules over Semisimple Lie Algebras}}{1009}{section.64.13}
\contentsline {subsection}{\numberline {64.13.1}\leavevmode {\color {Chapter }DominantWeights}}{1009}{subsection.64.13.1}
\contentsline {subsection}{\numberline {64.13.2}\leavevmode {\color {Chapter }DominantCharacter (for a semisimple Lie algebra and a highest weight)}}{1009}{subsection.64.13.2}
\contentsline {subsection}{\numberline {64.13.3}\leavevmode {\color {Chapter }DecomposeTensorProduct}}{1009}{subsection.64.13.3}
\contentsline {subsection}{\numberline {64.13.4}\leavevmode {\color {Chapter }DimensionOfHighestWeightModule}}{1010}{subsection.64.13.4}
\contentsline {section}{\numberline {64.14}\leavevmode {\color {Chapter }Admissible Lattices in UEA}}{1010}{section.64.14}
\contentsline {subsection}{\numberline {64.14.1}\leavevmode {\color {Chapter }IsUEALatticeElement}}{1011}{subsection.64.14.1}
\contentsline {subsection}{\numberline {64.14.2}\leavevmode {\color {Chapter }LatticeGeneratorsInUEA}}{1011}{subsection.64.14.2}
\contentsline {subsection}{\numberline {64.14.3}\leavevmode {\color {Chapter }ObjByExtRep (for creating a UEALattice element)}}{1011}{subsection.64.14.3}
\contentsline {subsection}{\numberline {64.14.4}\leavevmode {\color {Chapter }IsWeightRepElement}}{1012}{subsection.64.14.4}
\contentsline {subsection}{\numberline {64.14.5}\leavevmode {\color {Chapter }HighestWeightModule}}{1012}{subsection.64.14.5}
\contentsline {section}{\numberline {64.15}\leavevmode {\color {Chapter }Tensor Products and Exterior and Symmetric Powers}}{1013}{section.64.15}
\contentsline {subsection}{\numberline {64.15.1}\leavevmode {\color {Chapter }TensorProductOfAlgebraModules (for a list of algebra modules)}}{1013}{subsection.64.15.1}
\contentsline {subsection}{\numberline {64.15.2}\leavevmode {\color {Chapter }ExteriorPowerOfAlgebraModule}}{1013}{subsection.64.15.2}
\contentsline {subsection}{\numberline {64.15.3}\leavevmode {\color {Chapter }SymmetricPowerOfAlgebraModule}}{1014}{subsection.64.15.3}
\contentsline {chapter}{\numberline {65}\leavevmode {\color {Chapter }Magma Rings}}{1015}{chapter.65}
\contentsline {section}{\numberline {65.1}\leavevmode {\color {Chapter }Free Magma Rings}}{1016}{section.65.1}
\contentsline {subsection}{\numberline {65.1.1}\leavevmode {\color {Chapter }FreeMagmaRing}}{1016}{subsection.65.1.1}
\contentsline {subsection}{\numberline {65.1.2}\leavevmode {\color {Chapter }GroupRing}}{1016}{subsection.65.1.2}
\contentsline {subsection}{\numberline {65.1.3}\leavevmode {\color {Chapter }IsFreeMagmaRing}}{1016}{subsection.65.1.3}
\contentsline {subsection}{\numberline {65.1.4}\leavevmode {\color {Chapter }IsFreeMagmaRingWithOne}}{1017}{subsection.65.1.4}
\contentsline {subsection}{\numberline {65.1.5}\leavevmode {\color {Chapter }IsGroupRing}}{1017}{subsection.65.1.5}
\contentsline {subsection}{\numberline {65.1.6}\leavevmode {\color {Chapter }UnderlyingMagma}}{1017}{subsection.65.1.6}
\contentsline {subsection}{\numberline {65.1.7}\leavevmode {\color {Chapter }AugmentationIdeal}}{1017}{subsection.65.1.7}
\contentsline {section}{\numberline {65.2}\leavevmode {\color {Chapter }Elements of Free Magma Rings}}{1017}{section.65.2}
\contentsline {subsection}{\numberline {65.2.1}\leavevmode {\color {Chapter }IsMagmaRingObjDefaultRep}}{1017}{subsection.65.2.1}
\contentsline {subsection}{\numberline {65.2.2}\leavevmode {\color {Chapter }IsElementOfFreeMagmaRing}}{1018}{subsection.65.2.2}
\contentsline {subsection}{\numberline {65.2.3}\leavevmode {\color {Chapter }IsElementOfFreeMagmaRingFamily}}{1018}{subsection.65.2.3}
\contentsline {subsection}{\numberline {65.2.4}\leavevmode {\color {Chapter }CoefficientsAndMagmaElements}}{1018}{subsection.65.2.4}
\contentsline {subsection}{\numberline {65.2.5}\leavevmode {\color {Chapter }ZeroCoefficient}}{1018}{subsection.65.2.5}
\contentsline {subsection}{\numberline {65.2.6}\leavevmode {\color {Chapter }ElementOfMagmaRing}}{1018}{subsection.65.2.6}
\contentsline {section}{\numberline {65.3}\leavevmode {\color {Chapter }Natural Embeddings related to Magma Rings}}{1018}{section.65.3}
\contentsline {section}{\numberline {65.4}\leavevmode {\color {Chapter }Magma Rings modulo Relations}}{1019}{section.65.4}
\contentsline {subsection}{\numberline {65.4.1}\leavevmode {\color {Chapter }IsElementOfMagmaRingModuloRelations}}{1019}{subsection.65.4.1}
\contentsline {subsection}{\numberline {65.4.2}\leavevmode {\color {Chapter }IsElementOfMagmaRingModuloRelationsFamily}}{1019}{subsection.65.4.2}
\contentsline {subsection}{\numberline {65.4.3}\leavevmode {\color {Chapter }NormalizedElementOfMagmaRingModuloRelations}}{1020}{subsection.65.4.3}
\contentsline {subsection}{\numberline {65.4.4}\leavevmode {\color {Chapter }IsMagmaRingModuloRelations}}{1020}{subsection.65.4.4}
\contentsline {section}{\numberline {65.5}\leavevmode {\color {Chapter }Magma Rings modulo the Span of a Zero Element}}{1020}{section.65.5}
\contentsline {subsection}{\numberline {65.5.1}\leavevmode {\color {Chapter }IsElementOfMagmaRingModuloSpanOfZeroFamily}}{1020}{subsection.65.5.1}
\contentsline {subsection}{\numberline {65.5.2}\leavevmode {\color {Chapter }IsMagmaRingModuloSpanOfZero}}{1020}{subsection.65.5.2}
\contentsline {subsection}{\numberline {65.5.3}\leavevmode {\color {Chapter }MagmaRingModuloSpanOfZero}}{1020}{subsection.65.5.3}
\contentsline {section}{\numberline {65.6}\leavevmode {\color {Chapter }Technical Details about the Implementation of Magma Rings}}{1021}{section.65.6}
\contentsline {chapter}{\numberline {66}\leavevmode {\color {Chapter }Polynomials and Rational Functions}}{1022}{chapter.66}
\contentsline {section}{\numberline {66.1}\leavevmode {\color {Chapter }Indeterminates}}{1022}{section.66.1}
\contentsline {subsection}{\numberline {66.1.1}\leavevmode {\color {Chapter }Indeterminate}}{1023}{subsection.66.1.1}
\contentsline {subsection}{\numberline {66.1.2}\leavevmode {\color {Chapter }IndeterminateNumberOfUnivariateRationalFunction}}{1024}{subsection.66.1.2}
\contentsline {subsection}{\numberline {66.1.3}\leavevmode {\color {Chapter }IndeterminateOfUnivariateRationalFunction}}{1024}{subsection.66.1.3}
\contentsline {subsection}{\numberline {66.1.4}\leavevmode {\color {Chapter }IndeterminateName}}{1024}{subsection.66.1.4}
\contentsline {subsection}{\numberline {66.1.5}\leavevmode {\color {Chapter }CIUnivPols}}{1025}{subsection.66.1.5}
\contentsline {section}{\numberline {66.2}\leavevmode {\color {Chapter }Operations for Rational Functions}}{1025}{section.66.2}
\contentsline {section}{\numberline {66.3}\leavevmode {\color {Chapter }Comparison of Rational Functions}}{1026}{section.66.3}
\contentsline {section}{\numberline {66.4}\leavevmode {\color {Chapter }Properties and Attributes of Rational Functions}}{1027}{section.66.4}
\contentsline {subsection}{\numberline {66.4.1}\leavevmode {\color {Chapter }IsPolynomialFunction}}{1027}{subsection.66.4.1}
\contentsline {subsection}{\numberline {66.4.2}\leavevmode {\color {Chapter }NumeratorOfRationalFunction}}{1027}{subsection.66.4.2}
\contentsline {subsection}{\numberline {66.4.3}\leavevmode {\color {Chapter }DenominatorOfRationalFunction}}{1027}{subsection.66.4.3}
\contentsline {subsection}{\numberline {66.4.4}\leavevmode {\color {Chapter }IsPolynomial}}{1028}{subsection.66.4.4}
\contentsline {subsection}{\numberline {66.4.5}\leavevmode {\color {Chapter }AsPolynomial}}{1028}{subsection.66.4.5}
\contentsline {subsection}{\numberline {66.4.6}\leavevmode {\color {Chapter }IsUnivariateRationalFunction}}{1028}{subsection.66.4.6}
\contentsline {subsection}{\numberline {66.4.7}\leavevmode {\color {Chapter }CoefficientsOfUnivariateRationalFunction}}{1028}{subsection.66.4.7}
\contentsline {subsection}{\numberline {66.4.8}\leavevmode {\color {Chapter }IsUnivariatePolynomial}}{1029}{subsection.66.4.8}
\contentsline {subsection}{\numberline {66.4.9}\leavevmode {\color {Chapter }CoefficientsOfUnivariatePolynomial}}{1029}{subsection.66.4.9}
\contentsline {subsection}{\numberline {66.4.10}\leavevmode {\color {Chapter }IsLaurentPolynomial}}{1029}{subsection.66.4.10}
\contentsline {subsection}{\numberline {66.4.11}\leavevmode {\color {Chapter }IsConstantRationalFunction}}{1029}{subsection.66.4.11}
\contentsline {subsection}{\numberline {66.4.12}\leavevmode {\color {Chapter }IsPrimitivePolynomial}}{1029}{subsection.66.4.12}
\contentsline {subsection}{\numberline {66.4.13}\leavevmode {\color {Chapter }SplittingField}}{1029}{subsection.66.4.13}
\contentsline {section}{\numberline {66.5}\leavevmode {\color {Chapter }Univariate Polynomials}}{1030}{section.66.5}
\contentsline {subsection}{\numberline {66.5.1}\leavevmode {\color {Chapter }UnivariatePolynomial}}{1030}{subsection.66.5.1}
\contentsline {subsection}{\numberline {66.5.2}\leavevmode {\color {Chapter }UnivariatePolynomialByCoefficients}}{1030}{subsection.66.5.2}
\contentsline {subsection}{\numberline {66.5.3}\leavevmode {\color {Chapter }DegreeOfLaurentPolynomial}}{1030}{subsection.66.5.3}
\contentsline {subsection}{\numberline {66.5.4}\leavevmode {\color {Chapter }RootsOfPolynomial}}{1030}{subsection.66.5.4}
\contentsline {subsection}{\numberline {66.5.5}\leavevmode {\color {Chapter }RootsOfUPol}}{1031}{subsection.66.5.5}
\contentsline {subsection}{\numberline {66.5.6}\leavevmode {\color {Chapter }QuotRemLaurpols}}{1031}{subsection.66.5.6}
\contentsline {subsection}{\numberline {66.5.7}\leavevmode {\color {Chapter }UnivariatenessTestRationalFunction}}{1031}{subsection.66.5.7}
\contentsline {subsection}{\numberline {66.5.8}\leavevmode {\color {Chapter }InfoPoly}}{1032}{subsection.66.5.8}
\contentsline {section}{\numberline {66.6}\leavevmode {\color {Chapter }Polynomials as Univariate Polynomials in one Indeterminate}}{1032}{section.66.6}
\contentsline {subsection}{\numberline {66.6.1}\leavevmode {\color {Chapter }DegreeIndeterminate}}{1032}{subsection.66.6.1}
\contentsline {subsection}{\numberline {66.6.2}\leavevmode {\color {Chapter }PolynomialCoefficientsOfPolynomial}}{1032}{subsection.66.6.2}
\contentsline {subsection}{\numberline {66.6.3}\leavevmode {\color {Chapter }LeadingCoefficient}}{1033}{subsection.66.6.3}
\contentsline {subsection}{\numberline {66.6.4}\leavevmode {\color {Chapter }LeadingMonomial}}{1033}{subsection.66.6.4}
\contentsline {subsection}{\numberline {66.6.5}\leavevmode {\color {Chapter }Derivative}}{1033}{subsection.66.6.5}
\contentsline {subsection}{\numberline {66.6.6}\leavevmode {\color {Chapter }Discriminant}}{1033}{subsection.66.6.6}
\contentsline {subsection}{\numberline {66.6.7}\leavevmode {\color {Chapter }Resultant}}{1034}{subsection.66.6.7}
\contentsline {section}{\numberline {66.7}\leavevmode {\color {Chapter }Multivariate Polynomials}}{1034}{section.66.7}
\contentsline {subsection}{\numberline {66.7.1}\leavevmode {\color {Chapter }Value}}{1034}{subsection.66.7.1}
\contentsline {section}{\numberline {66.8}\leavevmode {\color {Chapter }Minimal Polynomials}}{1035}{section.66.8}
\contentsline {subsection}{\numberline {66.8.1}\leavevmode {\color {Chapter }MinimalPolynomial}}{1035}{subsection.66.8.1}
\contentsline {section}{\numberline {66.9}\leavevmode {\color {Chapter }Cyclotomic Polynomials}}{1035}{section.66.9}
\contentsline {subsection}{\numberline {66.9.1}\leavevmode {\color {Chapter }CyclotomicPolynomial}}{1035}{subsection.66.9.1}
\contentsline {section}{\numberline {66.10}\leavevmode {\color {Chapter }Polynomial Factorization}}{1035}{section.66.10}
\contentsline {subsection}{\numberline {66.10.1}\leavevmode {\color {Chapter }Factors (of polynomial)}}{1035}{subsection.66.10.1}
\contentsline {subsection}{\numberline {66.10.2}\leavevmode {\color {Chapter }FactorsSquarefree}}{1036}{subsection.66.10.2}
\contentsline {section}{\numberline {66.11}\leavevmode {\color {Chapter }Polynomials over the Rationals}}{1036}{section.66.11}
\contentsline {subsection}{\numberline {66.11.1}\leavevmode {\color {Chapter }PrimitivePolynomial}}{1036}{subsection.66.11.1}
\contentsline {subsection}{\numberline {66.11.2}\leavevmode {\color {Chapter }PolynomialModP}}{1036}{subsection.66.11.2}
\contentsline {subsection}{\numberline {66.11.3}\leavevmode {\color {Chapter }GaloisType}}{1037}{subsection.66.11.3}
\contentsline {subsection}{\numberline {66.11.4}\leavevmode {\color {Chapter }ProbabilityShapes}}{1037}{subsection.66.11.4}
\contentsline {section}{\numberline {66.12}\leavevmode {\color {Chapter }Factorization of Polynomials over the Rationals}}{1037}{section.66.12}
\contentsline {subsection}{\numberline {66.12.1}\leavevmode {\color {Chapter }BombieriNorm}}{1037}{subsection.66.12.1}
\contentsline {subsection}{\numberline {66.12.2}\leavevmode {\color {Chapter }MinimizedBombieriNorm}}{1038}{subsection.66.12.2}
\contentsline {subsection}{\numberline {66.12.3}\leavevmode {\color {Chapter }HenselBound}}{1038}{subsection.66.12.3}
\contentsline {subsection}{\numberline {66.12.4}\leavevmode {\color {Chapter }OneFactorBound}}{1038}{subsection.66.12.4}
\contentsline {section}{\numberline {66.13}\leavevmode {\color {Chapter }Laurent Polynomials}}{1038}{section.66.13}
\contentsline {subsection}{\numberline {66.13.1}\leavevmode {\color {Chapter }LaurentPolynomialByCoefficients}}{1038}{subsection.66.13.1}
\contentsline {subsection}{\numberline {66.13.2}\leavevmode {\color {Chapter }CoefficientsOfLaurentPolynomial}}{1038}{subsection.66.13.2}
\contentsline {subsection}{\numberline {66.13.3}\leavevmode {\color {Chapter }IndeterminateNumberOfLaurentPolynomial}}{1039}{subsection.66.13.3}
\contentsline {section}{\numberline {66.14}\leavevmode {\color {Chapter }Univariate Rational Functions}}{1039}{section.66.14}
\contentsline {subsection}{\numberline {66.14.1}\leavevmode {\color {Chapter }UnivariateRationalFunctionByCoefficients}}{1039}{subsection.66.14.1}
\contentsline {section}{\numberline {66.15}\leavevmode {\color {Chapter }Polynomial Rings and Function Fields}}{1039}{section.66.15}
\contentsline {subsection}{\numberline {66.15.1}\leavevmode {\color {Chapter }PolynomialRing}}{1040}{subsection.66.15.1}
\contentsline {subsection}{\numberline {66.15.2}\leavevmode {\color {Chapter }IndeterminatesOfPolynomialRing}}{1040}{subsection.66.15.2}
\contentsline {subsection}{\numberline {66.15.3}\leavevmode {\color {Chapter }CoefficientsRing}}{1040}{subsection.66.15.3}
\contentsline {subsection}{\numberline {66.15.4}\leavevmode {\color {Chapter }IsPolynomialRing}}{1041}{subsection.66.15.4}
\contentsline {subsection}{\numberline {66.15.5}\leavevmode {\color {Chapter }IsFiniteFieldPolynomialRing}}{1041}{subsection.66.15.5}
\contentsline {subsection}{\numberline {66.15.6}\leavevmode {\color {Chapter }IsAbelianNumberFieldPolynomialRing}}{1041}{subsection.66.15.6}
\contentsline {subsection}{\numberline {66.15.7}\leavevmode {\color {Chapter }IsRationalsPolynomialRing}}{1041}{subsection.66.15.7}
\contentsline {subsection}{\numberline {66.15.8}\leavevmode {\color {Chapter }FunctionField}}{1042}{subsection.66.15.8}
\contentsline {subsection}{\numberline {66.15.9}\leavevmode {\color {Chapter }IsFunctionField}}{1042}{subsection.66.15.9}
\contentsline {section}{\numberline {66.16}\leavevmode {\color {Chapter }Univariate Polynomial Rings}}{1042}{section.66.16}
\contentsline {subsection}{\numberline {66.16.1}\leavevmode {\color {Chapter }UnivariatePolynomialRing}}{1042}{subsection.66.16.1}
\contentsline {subsection}{\numberline {66.16.2}\leavevmode {\color {Chapter }IsUnivariatePolynomialRing}}{1042}{subsection.66.16.2}
\contentsline {section}{\numberline {66.17}\leavevmode {\color {Chapter }Monomial Orderings}}{1043}{section.66.17}
\contentsline {subsection}{\numberline {66.17.1}\leavevmode {\color {Chapter }IsMonomialOrdering}}{1043}{subsection.66.17.1}
\contentsline {subsection}{\numberline {66.17.2}\leavevmode {\color {Chapter }LeadingMonomialOfPolynomial}}{1043}{subsection.66.17.2}
\contentsline {subsection}{\numberline {66.17.3}\leavevmode {\color {Chapter }LeadingTermOfPolynomial}}{1044}{subsection.66.17.3}
\contentsline {subsection}{\numberline {66.17.4}\leavevmode {\color {Chapter }LeadingCoefficientOfPolynomial}}{1044}{subsection.66.17.4}
\contentsline {subsection}{\numberline {66.17.5}\leavevmode {\color {Chapter }MonomialComparisonFunction}}{1044}{subsection.66.17.5}
\contentsline {subsection}{\numberline {66.17.6}\leavevmode {\color {Chapter }MonomialExtrepComparisonFun}}{1044}{subsection.66.17.6}
\contentsline {subsection}{\numberline {66.17.7}\leavevmode {\color {Chapter }MonomialLexOrdering}}{1044}{subsection.66.17.7}
\contentsline {subsection}{\numberline {66.17.8}\leavevmode {\color {Chapter }MonomialGrlexOrdering}}{1045}{subsection.66.17.8}
\contentsline {subsection}{\numberline {66.17.9}\leavevmode {\color {Chapter }MonomialGrevlexOrdering}}{1045}{subsection.66.17.9}
\contentsline {subsection}{\numberline {66.17.10}\leavevmode {\color {Chapter }EliminationOrdering}}{1046}{subsection.66.17.10}
\contentsline {subsection}{\numberline {66.17.11}\leavevmode {\color {Chapter }PolynomialReduction}}{1046}{subsection.66.17.11}
\contentsline {subsection}{\numberline {66.17.12}\leavevmode {\color {Chapter }PolynomialReducedRemainder}}{1046}{subsection.66.17.12}
\contentsline {subsection}{\numberline {66.17.13}\leavevmode {\color {Chapter }PolynomialDivisionAlgorithm}}{1046}{subsection.66.17.13}
\contentsline {subsection}{\numberline {66.17.14}\leavevmode {\color {Chapter }MonomialExtGrlexLess}}{1047}{subsection.66.17.14}
\contentsline {section}{\numberline {66.18}\leavevmode {\color {Chapter }Groebner Bases}}{1047}{section.66.18}
\contentsline {subsection}{\numberline {66.18.1}\leavevmode {\color {Chapter }GroebnerBasis}}{1047}{subsection.66.18.1}
\contentsline {subsection}{\numberline {66.18.2}\leavevmode {\color {Chapter }ReducedGroebnerBasis}}{1048}{subsection.66.18.2}
\contentsline {subsection}{\numberline {66.18.3}\leavevmode {\color {Chapter }StoredGroebnerBasis}}{1048}{subsection.66.18.3}
\contentsline {subsection}{\numberline {66.18.4}\leavevmode {\color {Chapter }InfoGroebner}}{1048}{subsection.66.18.4}
\contentsline {section}{\numberline {66.19}\leavevmode {\color {Chapter }Rational Function Families}}{1048}{section.66.19}
\contentsline {subsection}{\numberline {66.19.1}\leavevmode {\color {Chapter }RationalFunctionsFamily}}{1049}{subsection.66.19.1}
\contentsline {subsection}{\numberline {66.19.2}\leavevmode {\color {Chapter }IsPolynomialFunctionsFamily}}{1049}{subsection.66.19.2}
\contentsline {subsection}{\numberline {66.19.3}\leavevmode {\color {Chapter }CoefficientsFamily}}{1049}{subsection.66.19.3}
\contentsline {section}{\numberline {66.20}\leavevmode {\color {Chapter }The Representations of Rational Functions}}{1049}{section.66.20}
\contentsline {section}{\numberline {66.21}\leavevmode {\color {Chapter }The Defining Attributes of Rational Functions}}{1050}{section.66.21}
\contentsline {subsection}{\numberline {66.21.1}\leavevmode {\color {Chapter }IsRationalFunctionDefaultRep}}{1051}{subsection.66.21.1}
\contentsline {subsection}{\numberline {66.21.2}\leavevmode {\color {Chapter }ExtRepNumeratorRatFun}}{1051}{subsection.66.21.2}
\contentsline {subsection}{\numberline {66.21.3}\leavevmode {\color {Chapter }ExtRepDenominatorRatFun}}{1051}{subsection.66.21.3}
\contentsline {subsection}{\numberline {66.21.4}\leavevmode {\color {Chapter }ZeroCoefficientRatFun}}{1051}{subsection.66.21.4}
\contentsline {subsection}{\numberline {66.21.5}\leavevmode {\color {Chapter }IsPolynomialDefaultRep}}{1051}{subsection.66.21.5}
\contentsline {subsection}{\numberline {66.21.6}\leavevmode {\color {Chapter }ExtRepPolynomialRatFun}}{1051}{subsection.66.21.6}
\contentsline {subsection}{\numberline {66.21.7}\leavevmode {\color {Chapter }IsLaurentPolynomialDefaultRep}}{1052}{subsection.66.21.7}
\contentsline {section}{\numberline {66.22}\leavevmode {\color {Chapter }Creation of Rational Functions}}{1052}{section.66.22}
\contentsline {subsection}{\numberline {66.22.1}\leavevmode {\color {Chapter }RationalFunctionByExtRep}}{1052}{subsection.66.22.1}
\contentsline {subsection}{\numberline {66.22.2}\leavevmode {\color {Chapter }PolynomialByExtRep}}{1052}{subsection.66.22.2}
\contentsline {subsection}{\numberline {66.22.3}\leavevmode {\color {Chapter }LaurentPolynomialByExtRep}}{1053}{subsection.66.22.3}
\contentsline {section}{\numberline {66.23}\leavevmode {\color {Chapter }Arithmetic for External Representations of Polynomials}}{1053}{section.66.23}
\contentsline {subsection}{\numberline {66.23.1}\leavevmode {\color {Chapter }ZippedSum}}{1053}{subsection.66.23.1}
\contentsline {subsection}{\numberline {66.23.2}\leavevmode {\color {Chapter }ZippedProduct}}{1053}{subsection.66.23.2}
\contentsline {subsection}{\numberline {66.23.3}\leavevmode {\color {Chapter }QuotientPolynomialsExtRep}}{1053}{subsection.66.23.3}
\contentsline {section}{\numberline {66.24}\leavevmode {\color {Chapter }Cancellation Tests for Rational Functions}}{1054}{section.66.24}
\contentsline {subsection}{\numberline {66.24.1}\leavevmode {\color {Chapter }RationalFunctionByExtRepWithCancellation}}{1054}{subsection.66.24.1}
\contentsline {subsection}{\numberline {66.24.2}\leavevmode {\color {Chapter }TryGcdCancelExtRepPolynomials}}{1054}{subsection.66.24.2}
\contentsline {subsection}{\numberline {66.24.3}\leavevmode {\color {Chapter }HeuristicCancelPolynomialsExtRep}}{1054}{subsection.66.24.3}
\contentsline {chapter}{\numberline {67}\leavevmode {\color {Chapter }Algebraic extensions of fields}}{1055}{chapter.67}
\contentsline {section}{\numberline {67.1}\leavevmode {\color {Chapter }Creation of Algebraic Extensions}}{1055}{section.67.1}
\contentsline {subsection}{\numberline {67.1.1}\leavevmode {\color {Chapter }AlgebraicExtension}}{1055}{subsection.67.1.1}
\contentsline {subsection}{\numberline {67.1.2}\leavevmode {\color {Chapter }IsAlgebraicExtension}}{1056}{subsection.67.1.2}
\contentsline {section}{\numberline {67.2}\leavevmode {\color {Chapter }Elements in Algebraic Extensions}}{1056}{section.67.2}
\contentsline {subsection}{\numberline {67.2.1}\leavevmode {\color {Chapter }IsAlgebraicElement}}{1057}{subsection.67.2.1}
\contentsline {section}{\numberline {67.3}\leavevmode {\color {Chapter }Finding Subfields}}{1057}{section.67.3}
\contentsline {subsection}{\numberline {67.3.1}\leavevmode {\color {Chapter }IdealDecompositionsOfPolynomial}}{1057}{subsection.67.3.1}
\contentsline {chapter}{\numberline {68}\leavevmode {\color {Chapter }p-adic Numbers (preliminary)}}{1058}{chapter.68}
\contentsline {section}{\numberline {68.1}\leavevmode {\color {Chapter }Pure p-adic Numbers}}{1058}{section.68.1}
\contentsline {subsection}{\numberline {68.1.1}\leavevmode {\color {Chapter }PurePadicNumberFamily}}{1058}{subsection.68.1.1}
\contentsline {subsection}{\numberline {68.1.2}\leavevmode {\color {Chapter }PadicNumber (for pure padics)}}{1058}{subsection.68.1.2}
\contentsline {subsection}{\numberline {68.1.3}\leavevmode {\color {Chapter }Valuation}}{1059}{subsection.68.1.3}
\contentsline {subsection}{\numberline {68.1.4}\leavevmode {\color {Chapter }ShiftedPadicNumber}}{1059}{subsection.68.1.4}
\contentsline {subsection}{\numberline {68.1.5}\leavevmode {\color {Chapter }IsPurePadicNumber}}{1059}{subsection.68.1.5}
\contentsline {subsection}{\numberline {68.1.6}\leavevmode {\color {Chapter }IsPurePadicNumberFamily}}{1059}{subsection.68.1.6}
\contentsline {section}{\numberline {68.2}\leavevmode {\color {Chapter }Extensions of the p-adic Numbers}}{1059}{section.68.2}
\contentsline {subsection}{\numberline {68.2.1}\leavevmode {\color {Chapter }PadicExtensionNumberFamily}}{1060}{subsection.68.2.1}
\contentsline {subsection}{\numberline {68.2.2}\leavevmode {\color {Chapter }PadicNumber (for a p-adic extension family and a rational)}}{1060}{subsection.68.2.2}
\contentsline {subsection}{\numberline {68.2.3}\leavevmode {\color {Chapter }IsPadicExtensionNumber}}{1061}{subsection.68.2.3}
\contentsline {subsection}{\numberline {68.2.4}\leavevmode {\color {Chapter }IsPadicExtensionNumberFamily}}{1061}{subsection.68.2.4}
\contentsline {chapter}{\numberline {69}\leavevmode {\color {Chapter }The MeatAxe}}{1062}{chapter.69}
\contentsline {section}{\numberline {69.1}\leavevmode {\color {Chapter }MeatAxe Modules}}{1062}{section.69.1}
\contentsline {subsection}{\numberline {69.1.1}\leavevmode {\color {Chapter }GModuleByMats}}{1062}{subsection.69.1.1}
\contentsline {section}{\numberline {69.2}\leavevmode {\color {Chapter }Module Constructions}}{1063}{section.69.2}
\contentsline {subsection}{\numberline {69.2.1}\leavevmode {\color {Chapter }PermutationGModule}}{1063}{subsection.69.2.1}
\contentsline {subsection}{\numberline {69.2.2}\leavevmode {\color {Chapter }TensorProductGModule}}{1063}{subsection.69.2.2}
\contentsline {subsection}{\numberline {69.2.3}\leavevmode {\color {Chapter }WedgeGModule}}{1063}{subsection.69.2.3}
\contentsline {section}{\numberline {69.3}\leavevmode {\color {Chapter }Selecting a Different MeatAxe}}{1063}{section.69.3}
\contentsline {subsection}{\numberline {69.3.1}\leavevmode {\color {Chapter }MTX}}{1063}{subsection.69.3.1}
\contentsline {section}{\numberline {69.4}\leavevmode {\color {Chapter }Accessing a Module}}{1063}{section.69.4}
\contentsline {subsection}{\numberline {69.4.1}\leavevmode {\color {Chapter }MTX.Generators}}{1064}{subsection.69.4.1}
\contentsline {subsection}{\numberline {69.4.2}\leavevmode {\color {Chapter }MTX.Dimension}}{1064}{subsection.69.4.2}
\contentsline {subsection}{\numberline {69.4.3}\leavevmode {\color {Chapter }MTX.Field}}{1064}{subsection.69.4.3}
\contentsline {section}{\numberline {69.5}\leavevmode {\color {Chapter }Irreducibility Tests}}{1064}{section.69.5}
\contentsline {subsection}{\numberline {69.5.1}\leavevmode {\color {Chapter }MTX.IsIrreducible}}{1064}{subsection.69.5.1}
\contentsline {subsection}{\numberline {69.5.2}\leavevmode {\color {Chapter }MTX.IsAbsolutelyIrreducible}}{1064}{subsection.69.5.2}
\contentsline {subsection}{\numberline {69.5.3}\leavevmode {\color {Chapter }MTX.DegreeSplittingField}}{1064}{subsection.69.5.3}
\contentsline {section}{\numberline {69.6}\leavevmode {\color {Chapter }Decomposition of modules}}{1064}{section.69.6}
\contentsline {subsection}{\numberline {69.6.1}\leavevmode {\color {Chapter }MTX.IsIndecomposable}}{1065}{subsection.69.6.1}
\contentsline {subsection}{\numberline {69.6.2}\leavevmode {\color {Chapter }MTX.Indecomposition}}{1065}{subsection.69.6.2}
\contentsline {subsection}{\numberline {69.6.3}\leavevmode {\color {Chapter }MTX.HomogeneousComponents}}{1065}{subsection.69.6.3}
\contentsline {section}{\numberline {69.7}\leavevmode {\color {Chapter }Finding Submodules}}{1065}{section.69.7}
\contentsline {subsection}{\numberline {69.7.1}\leavevmode {\color {Chapter }MTX.SubmoduleGModule}}{1065}{subsection.69.7.1}
\contentsline {subsection}{\numberline {69.7.2}\leavevmode {\color {Chapter }MTX.ProperSubmoduleBasis}}{1066}{subsection.69.7.2}
\contentsline {subsection}{\numberline {69.7.3}\leavevmode {\color {Chapter }MTX.BasesSubmodules}}{1066}{subsection.69.7.3}
\contentsline {subsection}{\numberline {69.7.4}\leavevmode {\color {Chapter }MTX.BasesMinimalSubmodules}}{1066}{subsection.69.7.4}
\contentsline {subsection}{\numberline {69.7.5}\leavevmode {\color {Chapter }MTX.BasesMaximalSubmodules}}{1066}{subsection.69.7.5}
\contentsline {subsection}{\numberline {69.7.6}\leavevmode {\color {Chapter }MTX.BasisRadical}}{1066}{subsection.69.7.6}
\contentsline {subsection}{\numberline {69.7.7}\leavevmode {\color {Chapter }MTX.BasisSocle}}{1066}{subsection.69.7.7}
\contentsline {subsection}{\numberline {69.7.8}\leavevmode {\color {Chapter }MTX.BasesMinimalSupermodules}}{1066}{subsection.69.7.8}
\contentsline {subsection}{\numberline {69.7.9}\leavevmode {\color {Chapter }MTX.BasesCompositionSeries}}{1066}{subsection.69.7.9}
\contentsline {subsection}{\numberline {69.7.10}\leavevmode {\color {Chapter }MTX.CompositionFactors}}{1067}{subsection.69.7.10}
\contentsline {subsection}{\numberline {69.7.11}\leavevmode {\color {Chapter }MTX.CollectedFactors}}{1067}{subsection.69.7.11}
\contentsline {section}{\numberline {69.8}\leavevmode {\color {Chapter }Induced Actions}}{1067}{section.69.8}
\contentsline {subsection}{\numberline {69.8.1}\leavevmode {\color {Chapter }MTX.NormedBasisAndBaseChange}}{1067}{subsection.69.8.1}
\contentsline {subsection}{\numberline {69.8.2}\leavevmode {\color {Chapter }MTX.InducedActionSubmodule}}{1067}{subsection.69.8.2}
\contentsline {subsection}{\numberline {69.8.3}\leavevmode {\color {Chapter }MTX.InducedActionFactorModule}}{1067}{subsection.69.8.3}
\contentsline {subsection}{\numberline {69.8.4}\leavevmode {\color {Chapter }MTX.InducedActionSubMatrix}}{1067}{subsection.69.8.4}
\contentsline {subsection}{\numberline {69.8.5}\leavevmode {\color {Chapter }MTX.InducedAction}}{1068}{subsection.69.8.5}
\contentsline {section}{\numberline {69.9}\leavevmode {\color {Chapter }Module Homomorphisms}}{1068}{section.69.9}
\contentsline {subsection}{\numberline {69.9.1}\leavevmode {\color {Chapter }MTX.BasisModuleHomomorphisms}}{1068}{subsection.69.9.1}
\contentsline {subsection}{\numberline {69.9.2}\leavevmode {\color {Chapter }MTX.BasisModuleEndomorphisms}}{1068}{subsection.69.9.2}
\contentsline {subsection}{\numberline {69.9.3}\leavevmode {\color {Chapter }MTX.IsomorphismModules}}{1068}{subsection.69.9.3}
\contentsline {subsection}{\numberline {69.9.4}\leavevmode {\color {Chapter }MTX.ModuleAutomorphisms}}{1068}{subsection.69.9.4}
\contentsline {section}{\numberline {69.10}\leavevmode {\color {Chapter }Module Homomorphisms for irreducible modules}}{1068}{section.69.10}
\contentsline {subsection}{\numberline {69.10.1}\leavevmode {\color {Chapter }MTX.IsEquivalent}}{1069}{subsection.69.10.1}
\contentsline {subsection}{\numberline {69.10.2}\leavevmode {\color {Chapter }MTX.IsomorphismIrred}}{1069}{subsection.69.10.2}
\contentsline {subsection}{\numberline {69.10.3}\leavevmode {\color {Chapter }MTX.Homomorphism}}{1069}{subsection.69.10.3}
\contentsline {subsection}{\numberline {69.10.4}\leavevmode {\color {Chapter }MTX.Homomorphisms}}{1069}{subsection.69.10.4}
\contentsline {subsection}{\numberline {69.10.5}\leavevmode {\color {Chapter }MTX.Distinguish}}{1069}{subsection.69.10.5}
\contentsline {section}{\numberline {69.11}\leavevmode {\color {Chapter }MeatAxe Functionality for Invariant Forms}}{1069}{section.69.11}
\contentsline {subsection}{\numberline {69.11.1}\leavevmode {\color {Chapter }MTX.InvariantBilinearForm}}{1069}{subsection.69.11.1}
\contentsline {subsection}{\numberline {69.11.2}\leavevmode {\color {Chapter }MTX.InvariantSesquilinearForm}}{1070}{subsection.69.11.2}
\contentsline {subsection}{\numberline {69.11.3}\leavevmode {\color {Chapter }MTX.InvariantQuadraticForm}}{1070}{subsection.69.11.3}
\contentsline {subsection}{\numberline {69.11.4}\leavevmode {\color {Chapter }MTX.BasisInOrbit}}{1070}{subsection.69.11.4}
\contentsline {subsection}{\numberline {69.11.5}\leavevmode {\color {Chapter }MTX.OrthogonalSign}}{1070}{subsection.69.11.5}
\contentsline {section}{\numberline {69.12}\leavevmode {\color {Chapter }The Smash MeatAxe}}{1070}{section.69.12}
\contentsline {subsection}{\numberline {69.12.1}\leavevmode {\color {Chapter }SMTX.RandomIrreducibleSubGModule}}{1070}{subsection.69.12.1}
\contentsline {subsection}{\numberline {69.12.2}\leavevmode {\color {Chapter }SMTX.GoodElementGModule}}{1071}{subsection.69.12.2}
\contentsline {subsection}{\numberline {69.12.3}\leavevmode {\color {Chapter }SMTX.SortHomGModule}}{1071}{subsection.69.12.3}
\contentsline {subsection}{\numberline {69.12.4}\leavevmode {\color {Chapter }SMTX.MinimalSubGModules}}{1071}{subsection.69.12.4}
\contentsline {subsection}{\numberline {69.12.5}\leavevmode {\color {Chapter }SMTX.Setter}}{1071}{subsection.69.12.5}
\contentsline {subsection}{\numberline {69.12.6}\leavevmode {\color {Chapter }SMTX.Getter}}{1071}{subsection.69.12.6}
\contentsline {subsection}{\numberline {69.12.7}\leavevmode {\color {Chapter }SMTX.IrreducibilityTest}}{1071}{subsection.69.12.7}
\contentsline {subsection}{\numberline {69.12.8}\leavevmode {\color {Chapter }SMTX.AbsoluteIrreducibilityTest}}{1071}{subsection.69.12.8}
\contentsline {subsection}{\numberline {69.12.9}\leavevmode {\color {Chapter }SMTX.MinimalSubGModule}}{1072}{subsection.69.12.9}
\contentsline {subsection}{\numberline {69.12.10}\leavevmode {\color {Chapter }SMTX.MatrixSum}}{1072}{subsection.69.12.10}
\contentsline {subsection}{\numberline {69.12.11}\leavevmode {\color {Chapter }SMTX.CompleteBasis}}{1072}{subsection.69.12.11}
\contentsline {section}{\numberline {69.13}\leavevmode {\color {Chapter }Smash MeatAxe Flags}}{1072}{section.69.13}
\contentsline {subsection}{\numberline {69.13.1}\leavevmode {\color {Chapter }SMTX.Subbasis}}{1072}{subsection.69.13.1}
\contentsline {subsection}{\numberline {69.13.2}\leavevmode {\color {Chapter }SMTX.AlgEl}}{1072}{subsection.69.13.2}
\contentsline {subsection}{\numberline {69.13.3}\leavevmode {\color {Chapter }SMTX.AlgElMat}}{1072}{subsection.69.13.3}
\contentsline {subsection}{\numberline {69.13.4}\leavevmode {\color {Chapter }SMTX.AlgElCharPol}}{1072}{subsection.69.13.4}
\contentsline {subsection}{\numberline {69.13.5}\leavevmode {\color {Chapter }SMTX.AlgElCharPolFac}}{1073}{subsection.69.13.5}
\contentsline {subsection}{\numberline {69.13.6}\leavevmode {\color {Chapter }SMTX.AlgElNullspaceVec}}{1073}{subsection.69.13.6}
\contentsline {subsection}{\numberline {69.13.7}\leavevmode {\color {Chapter }SMTX.AlgElNullspaceDimension}}{1073}{subsection.69.13.7}
\contentsline {subsection}{\numberline {69.13.8}\leavevmode {\color {Chapter }SMTX.CentMat}}{1073}{subsection.69.13.8}
\contentsline {subsection}{\numberline {69.13.9}\leavevmode {\color {Chapter }SMTX.CentMatMinPoly}}{1073}{subsection.69.13.9}
\contentsline {chapter}{\numberline {70}\leavevmode {\color {Chapter }Tables of Marks}}{1074}{chapter.70}
\contentsline {section}{\numberline {70.1}\leavevmode {\color {Chapter }More about Tables of Marks}}{1074}{section.70.1}
\contentsline {section}{\numberline {70.2}\leavevmode {\color {Chapter }Table of Marks Objects in GAP}}{1075}{section.70.2}
\contentsline {section}{\numberline {70.3}\leavevmode {\color {Chapter }Constructing Tables of Marks}}{1076}{section.70.3}
\contentsline {subsection}{\numberline {70.3.1}\leavevmode {\color {Chapter }TableOfMarks (for a group)}}{1076}{subsection.70.3.1}
\contentsline {subsection}{\numberline {70.3.2}\leavevmode {\color {Chapter }TableOfMarksByLattice}}{1077}{subsection.70.3.2}
\contentsline {subsection}{\numberline {70.3.3}\leavevmode {\color {Chapter }LatticeSubgroupsByTom}}{1077}{subsection.70.3.3}
\contentsline {section}{\numberline {70.4}\leavevmode {\color {Chapter }Printing Tables of Marks}}{1077}{section.70.4}
\contentsline {subsection}{\numberline {70.4.1}\leavevmode {\color {Chapter }ViewObj (for a table of marks)}}{1077}{subsection.70.4.1}
\contentsline {subsection}{\numberline {70.4.2}\leavevmode {\color {Chapter }PrintObj (for a table of marks)}}{1078}{subsection.70.4.2}
\contentsline {subsection}{\numberline {70.4.3}\leavevmode {\color {Chapter }Display (for a table of marks)}}{1078}{subsection.70.4.3}
\contentsline {section}{\numberline {70.5}\leavevmode {\color {Chapter }Sorting Tables of Marks}}{1079}{section.70.5}
\contentsline {subsection}{\numberline {70.5.1}\leavevmode {\color {Chapter }SortedTom}}{1079}{subsection.70.5.1}
\contentsline {subsection}{\numberline {70.5.2}\leavevmode {\color {Chapter }PermutationTom}}{1080}{subsection.70.5.2}
\contentsline {section}{\numberline {70.6}\leavevmode {\color {Chapter }Technical Details about Tables of Marks}}{1080}{section.70.6}
\contentsline {subsection}{\numberline {70.6.1}\leavevmode {\color {Chapter }InfoTom}}{1080}{subsection.70.6.1}
\contentsline {subsection}{\numberline {70.6.2}\leavevmode {\color {Chapter }IsTableOfMarks}}{1080}{subsection.70.6.2}
\contentsline {subsection}{\numberline {70.6.3}\leavevmode {\color {Chapter }TableOfMarksFamily}}{1080}{subsection.70.6.3}
\contentsline {subsection}{\numberline {70.6.4}\leavevmode {\color {Chapter }TableOfMarksComponents}}{1081}{subsection.70.6.4}
\contentsline {subsection}{\numberline {70.6.5}\leavevmode {\color {Chapter }ConvertToTableOfMarks}}{1081}{subsection.70.6.5}
\contentsline {section}{\numberline {70.7}\leavevmode {\color {Chapter }Attributes of Tables of Marks}}{1081}{section.70.7}
\contentsline {subsection}{\numberline {70.7.1}\leavevmode {\color {Chapter }MarksTom}}{1081}{subsection.70.7.1}
\contentsline {subsection}{\numberline {70.7.2}\leavevmode {\color {Chapter }NrSubsTom}}{1082}{subsection.70.7.2}
\contentsline {subsection}{\numberline {70.7.3}\leavevmode {\color {Chapter }LengthsTom}}{1082}{subsection.70.7.3}
\contentsline {subsection}{\numberline {70.7.4}\leavevmode {\color {Chapter }ClassTypesTom}}{1082}{subsection.70.7.4}
\contentsline {subsection}{\numberline {70.7.5}\leavevmode {\color {Chapter }ClassNamesTom}}{1083}{subsection.70.7.5}
\contentsline {subsection}{\numberline {70.7.6}\leavevmode {\color {Chapter }FusionsTom}}{1083}{subsection.70.7.6}
\contentsline {subsection}{\numberline {70.7.7}\leavevmode {\color {Chapter }UnderlyingGroup (for tables of marks)}}{1083}{subsection.70.7.7}
\contentsline {subsection}{\numberline {70.7.8}\leavevmode {\color {Chapter }IdempotentsTom}}{1084}{subsection.70.7.8}
\contentsline {subsection}{\numberline {70.7.9}\leavevmode {\color {Chapter }Identifier (for tables of marks)}}{1084}{subsection.70.7.9}
\contentsline {subsection}{\numberline {70.7.10}\leavevmode {\color {Chapter }MatTom}}{1084}{subsection.70.7.10}
\contentsline {subsection}{\numberline {70.7.11}\leavevmode {\color {Chapter }MoebiusTom}}{1085}{subsection.70.7.11}
\contentsline {subsection}{\numberline {70.7.12}\leavevmode {\color {Chapter }WeightsTom}}{1085}{subsection.70.7.12}
\contentsline {section}{\numberline {70.8}\leavevmode {\color {Chapter }Properties of Tables of Marks}}{1086}{section.70.8}
\contentsline {subsection}{\numberline {70.8.1}\leavevmode {\color {Chapter }IsAbelianTom}}{1086}{subsection.70.8.1}
\contentsline {section}{\numberline {70.9}\leavevmode {\color {Chapter }Other Operations for Tables of Marks}}{1086}{section.70.9}
\contentsline {subsection}{\numberline {70.9.1}\leavevmode {\color {Chapter }IsInternallyConsistent (for tables of marks)}}{1086}{subsection.70.9.1}
\contentsline {subsection}{\numberline {70.9.2}\leavevmode {\color {Chapter }DerivedSubgroupTom}}{1087}{subsection.70.9.2}
\contentsline {subsection}{\numberline {70.9.3}\leavevmode {\color {Chapter }DerivedSubgroupsTomPossible}}{1087}{subsection.70.9.3}
\contentsline {subsection}{\numberline {70.9.4}\leavevmode {\color {Chapter }NormalizerTom}}{1087}{subsection.70.9.4}
\contentsline {subsection}{\numberline {70.9.5}\leavevmode {\color {Chapter }ContainedTom}}{1088}{subsection.70.9.5}
\contentsline {subsection}{\numberline {70.9.6}\leavevmode {\color {Chapter }ContainingTom}}{1088}{subsection.70.9.6}
\contentsline {subsection}{\numberline {70.9.7}\leavevmode {\color {Chapter }CyclicExtensionsTom (for a prime)}}{1088}{subsection.70.9.7}
\contentsline {subsection}{\numberline {70.9.8}\leavevmode {\color {Chapter }DecomposedFixedPointVector}}{1089}{subsection.70.9.8}
\contentsline {subsection}{\numberline {70.9.9}\leavevmode {\color {Chapter }EulerianFunctionByTom}}{1089}{subsection.70.9.9}
\contentsline {subsection}{\numberline {70.9.10}\leavevmode {\color {Chapter }IntersectionsTom}}{1089}{subsection.70.9.10}
\contentsline {subsection}{\numberline {70.9.11}\leavevmode {\color {Chapter }FactorGroupTom}}{1090}{subsection.70.9.11}
\contentsline {subsection}{\numberline {70.9.12}\leavevmode {\color {Chapter }MaximalSubgroupsTom}}{1090}{subsection.70.9.12}
\contentsline {subsection}{\numberline {70.9.13}\leavevmode {\color {Chapter }MinimalSupergroupsTom}}{1091}{subsection.70.9.13}
\contentsline {section}{\numberline {70.10}\leavevmode {\color {Chapter }Accessing Subgroups via Tables of Marks}}{1091}{section.70.10}
\contentsline {subsection}{\numberline {70.10.1}\leavevmode {\color {Chapter }GeneratorsSubgroupsTom}}{1091}{subsection.70.10.1}
\contentsline {subsection}{\numberline {70.10.2}\leavevmode {\color {Chapter }StraightLineProgramsTom}}{1091}{subsection.70.10.2}
\contentsline {subsection}{\numberline {70.10.3}\leavevmode {\color {Chapter }IsTableOfMarksWithGens}}{1092}{subsection.70.10.3}
\contentsline {subsection}{\numberline {70.10.4}\leavevmode {\color {Chapter }RepresentativeTom}}{1093}{subsection.70.10.4}
\contentsline {section}{\numberline {70.11}\leavevmode {\color {Chapter }The Interface between Tables of Marks and Character Tables}}{1093}{section.70.11}
\contentsline {subsection}{\numberline {70.11.1}\leavevmode {\color {Chapter }FusionCharTableTom}}{1094}{subsection.70.11.1}
\contentsline {subsection}{\numberline {70.11.2}\leavevmode {\color {Chapter }PermCharsTom (via fusion map)}}{1094}{subsection.70.11.2}
\contentsline {section}{\numberline {70.12}\leavevmode {\color {Chapter }Generic Construction of Tables of Marks}}{1095}{section.70.12}
\contentsline {subsection}{\numberline {70.12.1}\leavevmode {\color {Chapter }TableOfMarksCyclic}}{1095}{subsection.70.12.1}
\contentsline {subsection}{\numberline {70.12.2}\leavevmode {\color {Chapter }TableOfMarksDihedral}}{1095}{subsection.70.12.2}
\contentsline {subsection}{\numberline {70.12.3}\leavevmode {\color {Chapter }TableOfMarksFrobenius}}{1096}{subsection.70.12.3}
\contentsline {section}{\numberline {70.13}\leavevmode {\color {Chapter }The Library of Tables of Marks}}{1096}{section.70.13}
\contentsline {chapter}{\numberline {71}\leavevmode {\color {Chapter }Character Tables}}{1097}{chapter.71}
\contentsline {section}{\numberline {71.1}\leavevmode {\color {Chapter }Some Remarks about Character Theory in \textsf {GAP}}}{1097}{section.71.1}
\contentsline {section}{\numberline {71.2}\leavevmode {\color {Chapter }History of Character Theory Stuff in GAP}}{1099}{section.71.2}
\contentsline {section}{\numberline {71.3}\leavevmode {\color {Chapter }Creating Character Tables}}{1100}{section.71.3}
\contentsline {subsection}{\numberline {71.3.1}\leavevmode {\color {Chapter }CharacterTable}}{1100}{subsection.71.3.1}
\contentsline {subsection}{\numberline {71.3.2}\leavevmode {\color {Chapter }BrauerTable}}{1101}{subsection.71.3.2}
\contentsline {subsection}{\numberline {71.3.3}\leavevmode {\color {Chapter }CharacterTableRegular}}{1102}{subsection.71.3.3}
\contentsline {subsection}{\numberline {71.3.4}\leavevmode {\color {Chapter }SupportedCharacterTableInfo}}{1102}{subsection.71.3.4}
\contentsline {subsection}{\numberline {71.3.5}\leavevmode {\color {Chapter }ConvertToCharacterTable}}{1103}{subsection.71.3.5}
\contentsline {section}{\numberline {71.4}\leavevmode {\color {Chapter }Character Table Categories}}{1103}{section.71.4}
\contentsline {subsection}{\numberline {71.4.1}\leavevmode {\color {Chapter }IsNearlyCharacterTable}}{1103}{subsection.71.4.1}
\contentsline {subsection}{\numberline {71.4.2}\leavevmode {\color {Chapter }InfoCharacterTable}}{1104}{subsection.71.4.2}
\contentsline {subsection}{\numberline {71.4.3}\leavevmode {\color {Chapter }NearlyCharacterTablesFamily}}{1104}{subsection.71.4.3}
\contentsline {section}{\numberline {71.5}\leavevmode {\color {Chapter }Conventions for Character Tables}}{1104}{section.71.5}
\contentsline {section}{\numberline {71.6}\leavevmode {\color {Chapter }The Interface between Character Tables and Groups}}{1105}{section.71.6}
\contentsline {subsection}{\numberline {71.6.1}\leavevmode {\color {Chapter }UnderlyingGroup (for character tables)}}{1105}{subsection.71.6.1}
\contentsline {subsection}{\numberline {71.6.2}\leavevmode {\color {Chapter }ConjugacyClasses (for character tables)}}{1105}{subsection.71.6.2}
\contentsline {subsection}{\numberline {71.6.3}\leavevmode {\color {Chapter }IdentificationOfConjugacyClasses}}{1106}{subsection.71.6.3}
\contentsline {subsection}{\numberline {71.6.4}\leavevmode {\color {Chapter }CharacterTableWithStoredGroup}}{1106}{subsection.71.6.4}
\contentsline {subsection}{\numberline {71.6.5}\leavevmode {\color {Chapter }CompatibleConjugacyClasses}}{1107}{subsection.71.6.5}
\contentsline {section}{\numberline {71.7}\leavevmode {\color {Chapter }Operators for Character Tables}}{1108}{section.71.7}
\contentsline {section}{\numberline {71.8}\leavevmode {\color {Chapter }Attributes and Properties for Groups and Character Tables}}{1108}{section.71.8}
\contentsline {subsection}{\numberline {71.8.1}\leavevmode {\color {Chapter }CharacterDegrees}}{1109}{subsection.71.8.1}
\contentsline {subsection}{\numberline {71.8.2}\leavevmode {\color {Chapter }Irr}}{1109}{subsection.71.8.2}
\contentsline {subsection}{\numberline {71.8.3}\leavevmode {\color {Chapter }LinearCharacters}}{1110}{subsection.71.8.3}
\contentsline {subsection}{\numberline {71.8.4}\leavevmode {\color {Chapter }OrdinaryCharacterTable}}{1110}{subsection.71.8.4}
\contentsline {subsection}{\numberline {71.8.5}\leavevmode {\color {Chapter }Group Operations Applicable to Character Tables}}{1111}{subsection.71.8.5}
\contentsline {section}{\numberline {71.9}\leavevmode {\color {Chapter }Attributes and Properties only for Character Tables}}{1112}{section.71.9}
\contentsline {subsection}{\numberline {71.9.1}\leavevmode {\color {Chapter }OrdersClassRepresentatives}}{1113}{subsection.71.9.1}
\contentsline {subsection}{\numberline {71.9.2}\leavevmode {\color {Chapter }SizesCentralizers}}{1113}{subsection.71.9.2}
\contentsline {subsection}{\numberline {71.9.3}\leavevmode {\color {Chapter }SizesConjugacyClasses}}{1113}{subsection.71.9.3}
\contentsline {subsection}{\numberline {71.9.4}\leavevmode {\color {Chapter }AutomorphismsOfTable}}{1113}{subsection.71.9.4}
\contentsline {subsection}{\numberline {71.9.5}\leavevmode {\color {Chapter }UnderlyingCharacteristic}}{1114}{subsection.71.9.5}
\contentsline {subsection}{\numberline {71.9.6}\leavevmode {\color {Chapter }Class Names and Character Names}}{1114}{subsection.71.9.6}
\contentsline {subsection}{\numberline {71.9.7}\leavevmode {\color {Chapter }Class Parameters and Character Parameters}}{1114}{subsection.71.9.7}
\contentsline {subsection}{\numberline {71.9.8}\leavevmode {\color {Chapter }Identifier (for character tables)}}{1115}{subsection.71.9.8}
\contentsline {subsection}{\numberline {71.9.9}\leavevmode {\color {Chapter }InfoText (for character tables)}}{1115}{subsection.71.9.9}
\contentsline {subsection}{\numberline {71.9.10}\leavevmode {\color {Chapter }InverseClasses}}{1116}{subsection.71.9.10}
\contentsline {subsection}{\numberline {71.9.11}\leavevmode {\color {Chapter }RealClasses}}{1116}{subsection.71.9.11}
\contentsline {subsection}{\numberline {71.9.12}\leavevmode {\color {Chapter }ClassOrbit}}{1116}{subsection.71.9.12}
\contentsline {subsection}{\numberline {71.9.13}\leavevmode {\color {Chapter }ClassRoots}}{1116}{subsection.71.9.13}
\contentsline {section}{\numberline {71.10}\leavevmode {\color {Chapter }Normal Subgroups Represented by Lists of Class Positions}}{1117}{section.71.10}
\contentsline {subsection}{\numberline {71.10.1}\leavevmode {\color {Chapter }ClassPositionsOfNormalSubgroups}}{1117}{subsection.71.10.1}
\contentsline {subsection}{\numberline {71.10.2}\leavevmode {\color {Chapter }ClassPositionsOfAgemo}}{1117}{subsection.71.10.2}
\contentsline {subsection}{\numberline {71.10.3}\leavevmode {\color {Chapter }ClassPositionsOfCentre (for a character table)}}{1117}{subsection.71.10.3}
\contentsline {subsection}{\numberline {71.10.4}\leavevmode {\color {Chapter }ClassPositionsOfDirectProductDecompositions}}{1118}{subsection.71.10.4}
\contentsline {subsection}{\numberline {71.10.5}\leavevmode {\color {Chapter }ClassPositionsOfDerivedSubgroup}}{1118}{subsection.71.10.5}
\contentsline {subsection}{\numberline {71.10.6}\leavevmode {\color {Chapter }ClassPositionsOfElementaryAbelianSeries}}{1118}{subsection.71.10.6}
\contentsline {subsection}{\numberline {71.10.7}\leavevmode {\color {Chapter }ClassPositionsOfFittingSubgroup}}{1118}{subsection.71.10.7}
\contentsline {subsection}{\numberline {71.10.8}\leavevmode {\color {Chapter }ClassPositionsOfLowerCentralSeries}}{1119}{subsection.71.10.8}
\contentsline {subsection}{\numberline {71.10.9}\leavevmode {\color {Chapter }ClassPositionsOfUpperCentralSeries}}{1119}{subsection.71.10.9}
\contentsline {subsection}{\numberline {71.10.10}\leavevmode {\color {Chapter }ClassPositionsOfSolvableRadical}}{1119}{subsection.71.10.10}
\contentsline {subsection}{\numberline {71.10.11}\leavevmode {\color {Chapter }ClassPositionsOfSupersolvableResiduum}}{1119}{subsection.71.10.11}
\contentsline {subsection}{\numberline {71.10.12}\leavevmode {\color {Chapter }ClassPositionsOfPCore}}{1120}{subsection.71.10.12}
\contentsline {subsection}{\numberline {71.10.13}\leavevmode {\color {Chapter }ClassPositionsOfNormalClosure}}{1120}{subsection.71.10.13}
\contentsline {section}{\numberline {71.11}\leavevmode {\color {Chapter }Operations Concerning Blocks}}{1120}{section.71.11}
\contentsline {subsection}{\numberline {71.11.1}\leavevmode {\color {Chapter }PrimeBlocks}}{1120}{subsection.71.11.1}
\contentsline {subsection}{\numberline {71.11.2}\leavevmode {\color {Chapter }SameBlock}}{1121}{subsection.71.11.2}
\contentsline {subsection}{\numberline {71.11.3}\leavevmode {\color {Chapter }BlocksInfo}}{1121}{subsection.71.11.3}
\contentsline {subsection}{\numberline {71.11.4}\leavevmode {\color {Chapter }DecompositionMatrix}}{1122}{subsection.71.11.4}
\contentsline {subsection}{\numberline {71.11.5}\leavevmode {\color {Chapter }LaTeXStringDecompositionMatrix}}{1123}{subsection.71.11.5}
\contentsline {section}{\numberline {71.12}\leavevmode {\color {Chapter }Other Operations for Character Tables}}{1124}{section.71.12}
\contentsline {subsection}{\numberline {71.12.1}\leavevmode {\color {Chapter }Index (for two character tables)}}{1124}{subsection.71.12.1}
\contentsline {subsection}{\numberline {71.12.2}\leavevmode {\color {Chapter }IsInternallyConsistent (for character tables)}}{1124}{subsection.71.12.2}
\contentsline {subsection}{\numberline {71.12.3}\leavevmode {\color {Chapter }IsPSolvableCharacterTable}}{1125}{subsection.71.12.3}
\contentsline {subsection}{\numberline {71.12.4}\leavevmode {\color {Chapter }IsClassFusionOfNormalSubgroup}}{1125}{subsection.71.12.4}
\contentsline {subsection}{\numberline {71.12.5}\leavevmode {\color {Chapter }Indicator}}{1125}{subsection.71.12.5}
\contentsline {subsection}{\numberline {71.12.6}\leavevmode {\color {Chapter }NrPolyhedralSubgroups}}{1126}{subsection.71.12.6}
\contentsline {subsection}{\numberline {71.12.7}\leavevmode {\color {Chapter }ClassMultiplicationCoefficient (for character tables)}}{1126}{subsection.71.12.7}
\contentsline {subsection}{\numberline {71.12.8}\leavevmode {\color {Chapter }ClassStructureCharTable}}{1127}{subsection.71.12.8}
\contentsline {subsection}{\numberline {71.12.9}\leavevmode {\color {Chapter }MatClassMultCoeffsCharTable}}{1127}{subsection.71.12.9}
\contentsline {section}{\numberline {71.13}\leavevmode {\color {Chapter }Printing Character Tables}}{1127}{section.71.13}
\contentsline {subsection}{\numberline {71.13.1}\leavevmode {\color {Chapter }ViewObj (for a character table)}}{1127}{subsection.71.13.1}
\contentsline {subsection}{\numberline {71.13.2}\leavevmode {\color {Chapter }PrintObj (for a character table)}}{1128}{subsection.71.13.2}
\contentsline {subsection}{\numberline {71.13.3}\leavevmode {\color {Chapter }Display (for a character table)}}{1128}{subsection.71.13.3}
\contentsline {subsection}{\numberline {71.13.4}\leavevmode {\color {Chapter }DisplayOptions}}{1130}{subsection.71.13.4}
\contentsline {subsection}{\numberline {71.13.5}\leavevmode {\color {Chapter }PrintCharacterTable}}{1131}{subsection.71.13.5}
\contentsline {section}{\numberline {71.14}\leavevmode {\color {Chapter }Computing the Irreducible Characters of a Group}}{1132}{section.71.14}
\contentsline {subsection}{\numberline {71.14.1}\leavevmode {\color {Chapter }IrrDixonSchneider}}{1132}{subsection.71.14.1}
\contentsline {subsection}{\numberline {71.14.2}\leavevmode {\color {Chapter }IrrConlon}}{1132}{subsection.71.14.2}
\contentsline {subsection}{\numberline {71.14.3}\leavevmode {\color {Chapter }IrrBaumClausen}}{1133}{subsection.71.14.3}
\contentsline {subsection}{\numberline {71.14.4}\leavevmode {\color {Chapter }IrreducibleRepresentations}}{1133}{subsection.71.14.4}
\contentsline {subsection}{\numberline {71.14.5}\leavevmode {\color {Chapter }IrreducibleRepresentationsDixon}}{1134}{subsection.71.14.5}
\contentsline {section}{\numberline {71.15}\leavevmode {\color {Chapter }Representations Given by Modules}}{1135}{section.71.15}
\contentsline {subsection}{\numberline {71.15.1}\leavevmode {\color {Chapter }IrreducibleModules}}{1135}{subsection.71.15.1}
\contentsline {subsection}{\numberline {71.15.2}\leavevmode {\color {Chapter }AbsolutelyIrreducibleModules}}{1135}{subsection.71.15.2}
\contentsline {subsection}{\numberline {71.15.3}\leavevmode {\color {Chapter }RegularModule}}{1135}{subsection.71.15.3}
\contentsline {section}{\numberline {71.16}\leavevmode {\color {Chapter }The Dixon-Schneider Algorithm}}{1135}{section.71.16}
\contentsline {section}{\numberline {71.17}\leavevmode {\color {Chapter }Advanced Methods for Dixon-Schneider Calculations}}{1136}{section.71.17}
\contentsline {subsection}{\numberline {71.17.1}\leavevmode {\color {Chapter }DixonRecord}}{1136}{subsection.71.17.1}
\contentsline {subsection}{\numberline {71.17.2}\leavevmode {\color {Chapter }DixonInit}}{1136}{subsection.71.17.2}
\contentsline {subsection}{\numberline {71.17.3}\leavevmode {\color {Chapter }DixontinI}}{1137}{subsection.71.17.3}
\contentsline {subsection}{\numberline {71.17.4}\leavevmode {\color {Chapter }DixonSplit}}{1137}{subsection.71.17.4}
\contentsline {subsection}{\numberline {71.17.5}\leavevmode {\color {Chapter }BestSplittingMatrix}}{1137}{subsection.71.17.5}
\contentsline {subsection}{\numberline {71.17.6}\leavevmode {\color {Chapter }DxIncludeIrreducibles}}{1137}{subsection.71.17.6}
\contentsline {subsection}{\numberline {71.17.7}\leavevmode {\color {Chapter }SplitCharacters}}{1138}{subsection.71.17.7}
\contentsline {subsection}{\numberline {71.17.8}\leavevmode {\color {Chapter }IsDxLargeGroup}}{1138}{subsection.71.17.8}
\contentsline {section}{\numberline {71.18}\leavevmode {\color {Chapter }Components of a Dixon Record}}{1138}{section.71.18}
\contentsline {section}{\numberline {71.19}\leavevmode {\color {Chapter }An Example of Advanced Dixon-Schneider Calculations}}{1138}{section.71.19}
\contentsline {section}{\numberline {71.20}\leavevmode {\color {Chapter }Constructing Character Tables from Others}}{1141}{section.71.20}
\contentsline {subsection}{\numberline {71.20.1}\leavevmode {\color {Chapter }CharacterTableDirectProduct}}{1141}{subsection.71.20.1}
\contentsline {subsection}{\numberline {71.20.2}\leavevmode {\color {Chapter }FactorsOfDirectProduct}}{1142}{subsection.71.20.2}
\contentsline {subsection}{\numberline {71.20.3}\leavevmode {\color {Chapter }CharacterTableFactorGroup}}{1142}{subsection.71.20.3}
\contentsline {subsection}{\numberline {71.20.4}\leavevmode {\color {Chapter }CharacterTableIsoclinic}}{1142}{subsection.71.20.4}
\contentsline {subsection}{\numberline {71.20.5}\leavevmode {\color {Chapter }CharacterTableOfNormalSubgroup}}{1145}{subsection.71.20.5}
\contentsline {subsection}{\numberline {71.20.6}\leavevmode {\color {Chapter }CharacterTableWreathSymmetric}}{1146}{subsection.71.20.6}
\contentsline {subsection}{\numberline {71.20.7}\leavevmode {\color {Chapter }CharacterValueWreathSymmetric}}{1146}{subsection.71.20.7}
\contentsline {section}{\numberline {71.21}\leavevmode {\color {Chapter }Sorted Character Tables}}{1147}{section.71.21}
\contentsline {subsection}{\numberline {71.21.1}\leavevmode {\color {Chapter }CharacterTableWithSortedCharacters}}{1147}{subsection.71.21.1}
\contentsline {subsection}{\numberline {71.21.2}\leavevmode {\color {Chapter }SortedCharacters}}{1147}{subsection.71.21.2}
\contentsline {subsection}{\numberline {71.21.3}\leavevmode {\color {Chapter }CharacterTableWithSortedClasses}}{1148}{subsection.71.21.3}
\contentsline {subsection}{\numberline {71.21.4}\leavevmode {\color {Chapter }SortedCharacterTable (w.r.t. a normal subgroup)}}{1148}{subsection.71.21.4}
\contentsline {subsection}{\numberline {71.21.5}\leavevmode {\color {Chapter }ClassPermutation}}{1149}{subsection.71.21.5}
\contentsline {section}{\numberline {71.22}\leavevmode {\color {Chapter }Automorphisms and Equivalence of Character Tables}}{1150}{section.71.22}
\contentsline {subsection}{\numberline {71.22.1}\leavevmode {\color {Chapter }MatrixAutomorphisms}}{1150}{subsection.71.22.1}
\contentsline {subsection}{\numberline {71.22.2}\leavevmode {\color {Chapter }TableAutomorphisms}}{1150}{subsection.71.22.2}
\contentsline {subsection}{\numberline {71.22.3}\leavevmode {\color {Chapter }TransformingPermutations}}{1151}{subsection.71.22.3}
\contentsline {subsection}{\numberline {71.22.4}\leavevmode {\color {Chapter }TransformingPermutationsCharacterTables}}{1151}{subsection.71.22.4}
\contentsline {subsection}{\numberline {71.22.5}\leavevmode {\color {Chapter }FamiliesOfRows}}{1152}{subsection.71.22.5}
\contentsline {section}{\numberline {71.23}\leavevmode {\color {Chapter }Storing Normal Subgroup Information}}{1152}{section.71.23}
\contentsline {subsection}{\numberline {71.23.1}\leavevmode {\color {Chapter }NormalSubgroupClassesInfo}}{1152}{subsection.71.23.1}
\contentsline {subsection}{\numberline {71.23.2}\leavevmode {\color {Chapter }ClassPositionsOfNormalSubgroup}}{1153}{subsection.71.23.2}
\contentsline {subsection}{\numberline {71.23.3}\leavevmode {\color {Chapter }NormalSubgroupClasses}}{1153}{subsection.71.23.3}
\contentsline {subsection}{\numberline {71.23.4}\leavevmode {\color {Chapter }FactorGroupNormalSubgroupClasses}}{1153}{subsection.71.23.4}
\contentsline {chapter}{\numberline {72}\leavevmode {\color {Chapter }Class Functions}}{1155}{chapter.72}
\contentsline {section}{\numberline {72.1}\leavevmode {\color {Chapter }Why Class Functions?}}{1155}{section.72.1}
\contentsline {subsection}{\numberline {72.1.1}\leavevmode {\color {Chapter }IsClassFunction}}{1157}{subsection.72.1.1}
\contentsline {section}{\numberline {72.2}\leavevmode {\color {Chapter }Basic Operations for Class Functions}}{1158}{section.72.2}
\contentsline {subsection}{\numberline {72.2.1}\leavevmode {\color {Chapter }UnderlyingCharacterTable}}{1158}{subsection.72.2.1}
\contentsline {subsection}{\numberline {72.2.2}\leavevmode {\color {Chapter }ValuesOfClassFunction}}{1158}{subsection.72.2.2}
\contentsline {section}{\numberline {72.3}\leavevmode {\color {Chapter }Comparison of Class Functions}}{1159}{section.72.3}
\contentsline {section}{\numberline {72.4}\leavevmode {\color {Chapter }Arithmetic Operations for Class Functions}}{1159}{section.72.4}
\contentsline {subsection}{\numberline {72.4.1}\leavevmode {\color {Chapter }Characteristic (for a class function)}}{1162}{subsection.72.4.1}
\contentsline {subsection}{\numberline {72.4.2}\leavevmode {\color {Chapter }ComplexConjugate (for a class function)}}{1162}{subsection.72.4.2}
\contentsline {subsection}{\numberline {72.4.3}\leavevmode {\color {Chapter }Order (for a class function)}}{1163}{subsection.72.4.3}
\contentsline {section}{\numberline {72.5}\leavevmode {\color {Chapter }Printing Class Functions}}{1163}{section.72.5}
\contentsline {subsection}{\numberline {72.5.1}\leavevmode {\color {Chapter }ViewObj (for class functions)}}{1163}{subsection.72.5.1}
\contentsline {subsection}{\numberline {72.5.2}\leavevmode {\color {Chapter }PrintObj (for class functions)}}{1163}{subsection.72.5.2}
\contentsline {subsection}{\numberline {72.5.3}\leavevmode {\color {Chapter }Display (for class functions)}}{1163}{subsection.72.5.3}
\contentsline {section}{\numberline {72.6}\leavevmode {\color {Chapter }Creating Class Functions from Values Lists}}{1164}{section.72.6}
\contentsline {subsection}{\numberline {72.6.1}\leavevmode {\color {Chapter }ClassFunction (for a character table and a list)}}{1164}{subsection.72.6.1}
\contentsline {subsection}{\numberline {72.6.2}\leavevmode {\color {Chapter }VirtualCharacter (for a character table and a list)}}{1164}{subsection.72.6.2}
\contentsline {subsection}{\numberline {72.6.3}\leavevmode {\color {Chapter }Character (for a character table and a list)}}{1164}{subsection.72.6.3}
\contentsline {subsection}{\numberline {72.6.4}\leavevmode {\color {Chapter }ClassFunctionSameType}}{1165}{subsection.72.6.4}
\contentsline {section}{\numberline {72.7}\leavevmode {\color {Chapter }Creating Class Functions using Groups}}{1165}{section.72.7}
\contentsline {subsection}{\numberline {72.7.1}\leavevmode {\color {Chapter }TrivialCharacter}}{1165}{subsection.72.7.1}
\contentsline {subsection}{\numberline {72.7.2}\leavevmode {\color {Chapter }NaturalCharacter (for a group)}}{1165}{subsection.72.7.2}
\contentsline {subsection}{\numberline {72.7.3}\leavevmode {\color {Chapter }PermutationCharacter}}{1166}{subsection.72.7.3}
\contentsline {section}{\numberline {72.8}\leavevmode {\color {Chapter }Operations for Class Functions}}{1166}{section.72.8}
\contentsline {subsection}{\numberline {72.8.1}\leavevmode {\color {Chapter }IsCharacter}}{1167}{subsection.72.8.1}
\contentsline {subsection}{\numberline {72.8.2}\leavevmode {\color {Chapter }IsVirtualCharacter}}{1167}{subsection.72.8.2}
\contentsline {subsection}{\numberline {72.8.3}\leavevmode {\color {Chapter }IsIrreducibleCharacter}}{1167}{subsection.72.8.3}
\contentsline {subsection}{\numberline {72.8.4}\leavevmode {\color {Chapter }DegreeOfCharacter}}{1167}{subsection.72.8.4}
\contentsline {subsection}{\numberline {72.8.5}\leavevmode {\color {Chapter }ScalarProduct (for characters)}}{1168}{subsection.72.8.5}
\contentsline {subsection}{\numberline {72.8.6}\leavevmode {\color {Chapter }MatScalarProducts}}{1168}{subsection.72.8.6}
\contentsline {subsection}{\numberline {72.8.7}\leavevmode {\color {Chapter }Norm (for a class function)}}{1168}{subsection.72.8.7}
\contentsline {subsection}{\numberline {72.8.8}\leavevmode {\color {Chapter }ConstituentsOfCharacter}}{1169}{subsection.72.8.8}
\contentsline {subsection}{\numberline {72.8.9}\leavevmode {\color {Chapter }KernelOfCharacter}}{1169}{subsection.72.8.9}
\contentsline {subsection}{\numberline {72.8.10}\leavevmode {\color {Chapter }ClassPositionsOfKernel}}{1169}{subsection.72.8.10}
\contentsline {subsection}{\numberline {72.8.11}\leavevmode {\color {Chapter }CentreOfCharacter}}{1170}{subsection.72.8.11}
\contentsline {subsection}{\numberline {72.8.12}\leavevmode {\color {Chapter }ClassPositionsOfCentre (for a character)}}{1170}{subsection.72.8.12}
\contentsline {subsection}{\numberline {72.8.13}\leavevmode {\color {Chapter }InertiaSubgroup}}{1170}{subsection.72.8.13}
\contentsline {subsection}{\numberline {72.8.14}\leavevmode {\color {Chapter }CycleStructureClass}}{1171}{subsection.72.8.14}
\contentsline {subsection}{\numberline {72.8.15}\leavevmode {\color {Chapter }IsTransitive (for a character)}}{1171}{subsection.72.8.15}
\contentsline {subsection}{\numberline {72.8.16}\leavevmode {\color {Chapter }Transitivity (for a character)}}{1171}{subsection.72.8.16}
\contentsline {subsection}{\numberline {72.8.17}\leavevmode {\color {Chapter }CentralCharacter}}{1171}{subsection.72.8.17}
\contentsline {subsection}{\numberline {72.8.18}\leavevmode {\color {Chapter }DeterminantOfCharacter}}{1171}{subsection.72.8.18}
\contentsline {subsection}{\numberline {72.8.19}\leavevmode {\color {Chapter }EigenvaluesChar}}{1172}{subsection.72.8.19}
\contentsline {subsection}{\numberline {72.8.20}\leavevmode {\color {Chapter }Tensored}}{1172}{subsection.72.8.20}
\contentsline {section}{\numberline {72.9}\leavevmode {\color {Chapter }Restricted and Induced Class Functions}}{1173}{section.72.9}
\contentsline {subsection}{\numberline {72.9.1}\leavevmode {\color {Chapter }RestrictedClassFunction}}{1173}{subsection.72.9.1}
\contentsline {subsection}{\numberline {72.9.2}\leavevmode {\color {Chapter }RestrictedClassFunctions}}{1173}{subsection.72.9.2}
\contentsline {subsection}{\numberline {72.9.3}\leavevmode {\color {Chapter }InducedClassFunction}}{1174}{subsection.72.9.3}
\contentsline {subsection}{\numberline {72.9.4}\leavevmode {\color {Chapter }InducedClassFunctions}}{1174}{subsection.72.9.4}
\contentsline {subsection}{\numberline {72.9.5}\leavevmode {\color {Chapter }InducedClassFunctionsByFusionMap}}{1174}{subsection.72.9.5}
\contentsline {subsection}{\numberline {72.9.6}\leavevmode {\color {Chapter }InducedCyclic}}{1175}{subsection.72.9.6}
\contentsline {section}{\numberline {72.10}\leavevmode {\color {Chapter }Reducing Virtual Characters}}{1175}{section.72.10}
\contentsline {subsection}{\numberline {72.10.1}\leavevmode {\color {Chapter }ReducedClassFunctions}}{1175}{subsection.72.10.1}
\contentsline {subsection}{\numberline {72.10.2}\leavevmode {\color {Chapter }ReducedCharacters}}{1176}{subsection.72.10.2}
\contentsline {subsection}{\numberline {72.10.3}\leavevmode {\color {Chapter }IrreducibleDifferences}}{1176}{subsection.72.10.3}
\contentsline {subsection}{\numberline {72.10.4}\leavevmode {\color {Chapter }LLL}}{1176}{subsection.72.10.4}
\contentsline {subsection}{\numberline {72.10.5}\leavevmode {\color {Chapter }Extract}}{1177}{subsection.72.10.5}
\contentsline {subsection}{\numberline {72.10.6}\leavevmode {\color {Chapter }OrthogonalEmbeddingsSpecialDimension}}{1178}{subsection.72.10.6}
\contentsline {subsection}{\numberline {72.10.7}\leavevmode {\color {Chapter }Decreased}}{1180}{subsection.72.10.7}
\contentsline {subsection}{\numberline {72.10.8}\leavevmode {\color {Chapter }DnLattice}}{1181}{subsection.72.10.8}
\contentsline {subsection}{\numberline {72.10.9}\leavevmode {\color {Chapter }DnLatticeIterative}}{1182}{subsection.72.10.9}
\contentsline {section}{\numberline {72.11}\leavevmode {\color {Chapter }Symmetrizations of Class Functions}}{1182}{section.72.11}
\contentsline {subsection}{\numberline {72.11.1}\leavevmode {\color {Chapter }Symmetrizations}}{1182}{subsection.72.11.1}
\contentsline {subsection}{\numberline {72.11.2}\leavevmode {\color {Chapter }SymmetricParts}}{1183}{subsection.72.11.2}
\contentsline {subsection}{\numberline {72.11.3}\leavevmode {\color {Chapter }AntiSymmetricParts}}{1183}{subsection.72.11.3}
\contentsline {subsection}{\numberline {72.11.4}\leavevmode {\color {Chapter }OrthogonalComponents}}{1183}{subsection.72.11.4}
\contentsline {subsection}{\numberline {72.11.5}\leavevmode {\color {Chapter }SymplecticComponents}}{1184}{subsection.72.11.5}
\contentsline {section}{\numberline {72.12}\leavevmode {\color {Chapter }Molien Series}}{1185}{section.72.12}
\contentsline {subsection}{\numberline {72.12.1}\leavevmode {\color {Chapter }MolienSeries}}{1185}{subsection.72.12.1}
\contentsline {subsection}{\numberline {72.12.2}\leavevmode {\color {Chapter }MolienSeriesInfo}}{1185}{subsection.72.12.2}
\contentsline {subsection}{\numberline {72.12.3}\leavevmode {\color {Chapter }ValueMolienSeries}}{1186}{subsection.72.12.3}
\contentsline {subsection}{\numberline {72.12.4}\leavevmode {\color {Chapter }MolienSeriesWithGivenDenominator}}{1187}{subsection.72.12.4}
\contentsline {section}{\numberline {72.13}\leavevmode {\color {Chapter }Possible Permutation Characters}}{1187}{section.72.13}
\contentsline {subsection}{\numberline {72.13.1}\leavevmode {\color {Chapter }PermCharInfo}}{1188}{subsection.72.13.1}
\contentsline {subsection}{\numberline {72.13.2}\leavevmode {\color {Chapter }PermCharInfoRelative}}{1189}{subsection.72.13.2}
\contentsline {section}{\numberline {72.14}\leavevmode {\color {Chapter }Computing Possible Permutation Characters}}{1190}{section.72.14}
\contentsline {subsection}{\numberline {72.14.1}\leavevmode {\color {Chapter }PermChars}}{1190}{subsection.72.14.1}
\contentsline {subsection}{\numberline {72.14.2}\leavevmode {\color {Chapter }TestPerm1, ..., TestPerm5}}{1192}{subsection.72.14.2}
\contentsline {subsection}{\numberline {72.14.3}\leavevmode {\color {Chapter }PermBounds}}{1193}{subsection.72.14.3}
\contentsline {subsection}{\numberline {72.14.4}\leavevmode {\color {Chapter }PermComb}}{1194}{subsection.72.14.4}
\contentsline {subsection}{\numberline {72.14.5}\leavevmode {\color {Chapter }Inequalities}}{1194}{subsection.72.14.5}
\contentsline {section}{\numberline {72.15}\leavevmode {\color {Chapter }Operations for Brauer Characters}}{1195}{section.72.15}
\contentsline {subsection}{\numberline {72.15.1}\leavevmode {\color {Chapter }FrobeniusCharacterValue}}{1195}{subsection.72.15.1}
\contentsline {subsection}{\numberline {72.15.2}\leavevmode {\color {Chapter }BrauerCharacterValue}}{1195}{subsection.72.15.2}
\contentsline {subsection}{\numberline {72.15.3}\leavevmode {\color {Chapter }SizeOfFieldOfDefinition}}{1196}{subsection.72.15.3}
\contentsline {subsection}{\numberline {72.15.4}\leavevmode {\color {Chapter }RealizableBrauerCharacters}}{1196}{subsection.72.15.4}
\contentsline {section}{\numberline {72.16}\leavevmode {\color {Chapter }Domains Generated by Class Functions}}{1196}{section.72.16}
\contentsline {chapter}{\numberline {73}\leavevmode {\color {Chapter }Maps Concerning Character Tables}}{1197}{chapter.73}
\contentsline {section}{\numberline {73.1}\leavevmode {\color {Chapter }Power Maps}}{1197}{section.73.1}
\contentsline {subsection}{\numberline {73.1.1}\leavevmode {\color {Chapter }PowerMap}}{1198}{subsection.73.1.1}
\contentsline {subsection}{\numberline {73.1.2}\leavevmode {\color {Chapter }PossiblePowerMaps}}{1199}{subsection.73.1.2}
\contentsline {subsection}{\numberline {73.1.3}\leavevmode {\color {Chapter }ElementOrdersPowerMap}}{1200}{subsection.73.1.3}
\contentsline {subsection}{\numberline {73.1.4}\leavevmode {\color {Chapter }PowerMapByComposition}}{1201}{subsection.73.1.4}
\contentsline {section}{\numberline {73.2}\leavevmode {\color {Chapter }Orbits on Sets of Possible Power Maps}}{1201}{section.73.2}
\contentsline {subsection}{\numberline {73.2.1}\leavevmode {\color {Chapter }OrbitPowerMaps}}{1201}{subsection.73.2.1}
\contentsline {subsection}{\numberline {73.2.2}\leavevmode {\color {Chapter }RepresentativesPowerMaps}}{1201}{subsection.73.2.2}
\contentsline {section}{\numberline {73.3}\leavevmode {\color {Chapter }Class Fusions between Character Tables}}{1202}{section.73.3}
\contentsline {subsection}{\numberline {73.3.1}\leavevmode {\color {Chapter }FusionConjugacyClasses}}{1203}{subsection.73.3.1}
\contentsline {subsection}{\numberline {73.3.2}\leavevmode {\color {Chapter }ComputedClassFusions}}{1204}{subsection.73.3.2}
\contentsline {subsection}{\numberline {73.3.3}\leavevmode {\color {Chapter }GetFusionMap}}{1204}{subsection.73.3.3}
\contentsline {subsection}{\numberline {73.3.4}\leavevmode {\color {Chapter }StoreFusion}}{1205}{subsection.73.3.4}
\contentsline {subsection}{\numberline {73.3.5}\leavevmode {\color {Chapter }NamesOfFusionSources}}{1205}{subsection.73.3.5}
\contentsline {subsection}{\numberline {73.3.6}\leavevmode {\color {Chapter }PossibleClassFusions}}{1206}{subsection.73.3.6}
\contentsline {subsection}{\numberline {73.3.7}\leavevmode {\color {Chapter }ConsiderStructureConstants}}{1207}{subsection.73.3.7}
\contentsline {section}{\numberline {73.4}\leavevmode {\color {Chapter }Orbits on Sets of Possible Class Fusions}}{1208}{section.73.4}
\contentsline {subsection}{\numberline {73.4.1}\leavevmode {\color {Chapter }OrbitFusions}}{1208}{subsection.73.4.1}
\contentsline {subsection}{\numberline {73.4.2}\leavevmode {\color {Chapter }RepresentativesFusions}}{1208}{subsection.73.4.2}
\contentsline {section}{\numberline {73.5}\leavevmode {\color {Chapter }Parametrized Maps}}{1208}{section.73.5}
\contentsline {subsection}{\numberline {73.5.1}\leavevmode {\color {Chapter }CompositionMaps}}{1209}{subsection.73.5.1}
\contentsline {subsection}{\numberline {73.5.2}\leavevmode {\color {Chapter }InverseMap}}{1209}{subsection.73.5.2}
\contentsline {subsection}{\numberline {73.5.3}\leavevmode {\color {Chapter }ProjectionMap}}{1210}{subsection.73.5.3}
\contentsline {subsection}{\numberline {73.5.4}\leavevmode {\color {Chapter }Indirected}}{1210}{subsection.73.5.4}
\contentsline {subsection}{\numberline {73.5.5}\leavevmode {\color {Chapter }Parametrized}}{1210}{subsection.73.5.5}
\contentsline {subsection}{\numberline {73.5.6}\leavevmode {\color {Chapter }ContainedMaps}}{1211}{subsection.73.5.6}
\contentsline {subsection}{\numberline {73.5.7}\leavevmode {\color {Chapter }UpdateMap}}{1211}{subsection.73.5.7}
\contentsline {subsection}{\numberline {73.5.8}\leavevmode {\color {Chapter }MeetMaps}}{1212}{subsection.73.5.8}
\contentsline {subsection}{\numberline {73.5.9}\leavevmode {\color {Chapter }CommutativeDiagram}}{1212}{subsection.73.5.9}
\contentsline {subsection}{\numberline {73.5.10}\leavevmode {\color {Chapter }CheckFixedPoints}}{1213}{subsection.73.5.10}
\contentsline {subsection}{\numberline {73.5.11}\leavevmode {\color {Chapter }TransferDiagram}}{1213}{subsection.73.5.11}
\contentsline {subsection}{\numberline {73.5.12}\leavevmode {\color {Chapter }TestConsistencyMaps}}{1214}{subsection.73.5.12}
\contentsline {subsection}{\numberline {73.5.13}\leavevmode {\color {Chapter }Indeterminateness}}{1215}{subsection.73.5.13}
\contentsline {subsection}{\numberline {73.5.14}\leavevmode {\color {Chapter }PrintAmbiguity}}{1215}{subsection.73.5.14}
\contentsline {subsection}{\numberline {73.5.15}\leavevmode {\color {Chapter }ContainedSpecialVectors}}{1215}{subsection.73.5.15}
\contentsline {subsection}{\numberline {73.5.16}\leavevmode {\color {Chapter }CollapsedMat}}{1216}{subsection.73.5.16}
\contentsline {subsection}{\numberline {73.5.17}\leavevmode {\color {Chapter }ContainedDecomposables}}{1217}{subsection.73.5.17}
\contentsline {section}{\numberline {73.6}\leavevmode {\color {Chapter }Subroutines for the Construction of Power Maps}}{1218}{section.73.6}
\contentsline {subsection}{\numberline {73.6.1}\leavevmode {\color {Chapter }InitPowerMap}}{1218}{subsection.73.6.1}
\contentsline {subsection}{\numberline {73.6.2}\leavevmode {\color {Chapter }Congruences (for character tables)}}{1218}{subsection.73.6.2}
\contentsline {subsection}{\numberline {73.6.3}\leavevmode {\color {Chapter }ConsiderKernels}}{1219}{subsection.73.6.3}
\contentsline {subsection}{\numberline {73.6.4}\leavevmode {\color {Chapter }ConsiderSmallerPowerMaps}}{1219}{subsection.73.6.4}
\contentsline {subsection}{\numberline {73.6.5}\leavevmode {\color {Chapter }MinusCharacter}}{1219}{subsection.73.6.5}
\contentsline {subsection}{\numberline {73.6.6}\leavevmode {\color {Chapter }PowerMapsAllowedBySymmetrizations}}{1220}{subsection.73.6.6}
\contentsline {section}{\numberline {73.7}\leavevmode {\color {Chapter }Subroutines for the Construction of Class Fusions}}{1221}{section.73.7}
\contentsline {subsection}{\numberline {73.7.1}\leavevmode {\color {Chapter }InitFusion}}{1221}{subsection.73.7.1}
\contentsline {subsection}{\numberline {73.7.2}\leavevmode {\color {Chapter }CheckPermChar}}{1221}{subsection.73.7.2}
\contentsline {subsection}{\numberline {73.7.3}\leavevmode {\color {Chapter }ConsiderTableAutomorphisms}}{1222}{subsection.73.7.3}
\contentsline {subsection}{\numberline {73.7.4}\leavevmode {\color {Chapter }FusionsAllowedByRestrictions}}{1222}{subsection.73.7.4}
\contentsline {chapter}{\numberline {74}\leavevmode {\color {Chapter }Unknowns}}{1224}{chapter.74}
\contentsline {section}{\numberline {74.1}\leavevmode {\color {Chapter }More about Unknowns}}{1224}{section.74.1}
\contentsline {subsection}{\numberline {74.1.1}\leavevmode {\color {Chapter }Unknown}}{1224}{subsection.74.1.1}
\contentsline {subsection}{\numberline {74.1.2}\leavevmode {\color {Chapter }LargestUnknown}}{1225}{subsection.74.1.2}
\contentsline {subsection}{\numberline {74.1.3}\leavevmode {\color {Chapter }IsUnknown}}{1225}{subsection.74.1.3}
\contentsline {subsection}{\numberline {74.1.4}\leavevmode {\color {Chapter }Comparison of Unknowns}}{1225}{subsection.74.1.4}
\contentsline {subsection}{\numberline {74.1.5}\leavevmode {\color {Chapter }Arithmetical Operations for Unknowns}}{1225}{subsection.74.1.5}
\contentsline {chapter}{\numberline {75}\leavevmode {\color {Chapter }Monomiality Questions}}{1227}{chapter.75}
\contentsline {section}{\numberline {75.1}\leavevmode {\color {Chapter }InfoMonomial (Info Class)}}{1228}{section.75.1}
\contentsline {subsection}{\numberline {75.1.1}\leavevmode {\color {Chapter }InfoMonomial}}{1228}{subsection.75.1.1}
\contentsline {section}{\numberline {75.2}\leavevmode {\color {Chapter }Character Degrees and Derived Length}}{1228}{section.75.2}
\contentsline {subsection}{\numberline {75.2.1}\leavevmode {\color {Chapter }Alpha}}{1228}{subsection.75.2.1}
\contentsline {subsection}{\numberline {75.2.2}\leavevmode {\color {Chapter }Delta}}{1228}{subsection.75.2.2}
\contentsline {subsection}{\numberline {75.2.3}\leavevmode {\color {Chapter }IsBergerCondition}}{1228}{subsection.75.2.3}
\contentsline {section}{\numberline {75.3}\leavevmode {\color {Chapter }Primitivity of Characters}}{1229}{section.75.3}
\contentsline {subsection}{\numberline {75.3.1}\leavevmode {\color {Chapter }TestHomogeneous}}{1229}{subsection.75.3.1}
\contentsline {subsection}{\numberline {75.3.2}\leavevmode {\color {Chapter }IsPrimitiveCharacter}}{1230}{subsection.75.3.2}
\contentsline {subsection}{\numberline {75.3.3}\leavevmode {\color {Chapter }TestQuasiPrimitive}}{1230}{subsection.75.3.3}
\contentsline {subsection}{\numberline {75.3.4}\leavevmode {\color {Chapter }TestInducedFromNormalSubgroup}}{1231}{subsection.75.3.4}
\contentsline {section}{\numberline {75.4}\leavevmode {\color {Chapter }Testing Monomiality}}{1231}{section.75.4}
\contentsline {subsection}{\numberline {75.4.1}\leavevmode {\color {Chapter }TestMonomial}}{1231}{subsection.75.4.1}
\contentsline {subsection}{\numberline {75.4.2}\leavevmode {\color {Chapter }TestMonomialUseLattice}}{1232}{subsection.75.4.2}
\contentsline {subsection}{\numberline {75.4.3}\leavevmode {\color {Chapter }IsMonomialNumber}}{1232}{subsection.75.4.3}
\contentsline {subsection}{\numberline {75.4.4}\leavevmode {\color {Chapter }TestMonomialQuick}}{1233}{subsection.75.4.4}
\contentsline {subsection}{\numberline {75.4.5}\leavevmode {\color {Chapter }TestSubnormallyMonomial}}{1233}{subsection.75.4.5}
\contentsline {subsection}{\numberline {75.4.6}\leavevmode {\color {Chapter }TestRelativelySM}}{1234}{subsection.75.4.6}
\contentsline {section}{\numberline {75.5}\leavevmode {\color {Chapter }Minimal Nonmonomial Groups}}{1235}{section.75.5}
\contentsline {subsection}{\numberline {75.5.1}\leavevmode {\color {Chapter }IsMinimalNonmonomial}}{1235}{subsection.75.5.1}
\contentsline {subsection}{\numberline {75.5.2}\leavevmode {\color {Chapter }MinimalNonmonomialGroup}}{1235}{subsection.75.5.2}
\contentsline {chapter}{\numberline {76}\leavevmode {\color {Chapter }Using and Developing GAP Packages}}{1236}{chapter.76}
\contentsline {section}{\numberline {76.1}\leavevmode {\color {Chapter }Installing a GAP Package}}{1236}{section.76.1}
\contentsline {section}{\numberline {76.2}\leavevmode {\color {Chapter }Loading a GAP Package}}{1237}{section.76.2}
\contentsline {subsection}{\numberline {76.2.1}\leavevmode {\color {Chapter }LoadPackage}}{1237}{subsection.76.2.1}
\contentsline {subsection}{\numberline {76.2.2}\leavevmode {\color {Chapter }Automatic loading of \textsf {GAP} packages}}{1238}{subsection.76.2.2}
\contentsline {subsection}{\numberline {76.2.3}\leavevmode {\color {Chapter }SetPackagePath}}{1238}{subsection.76.2.3}
\contentsline {subsection}{\numberline {76.2.4}\leavevmode {\color {Chapter }ExtendRootDirectories}}{1238}{subsection.76.2.4}
\contentsline {subsection}{\numberline {76.2.5}\leavevmode {\color {Chapter }DisplayPackageLoadingLog}}{1239}{subsection.76.2.5}
\contentsline {section}{\numberline {76.3}\leavevmode {\color {Chapter }Functions for GAP Packages}}{1240}{section.76.3}
\contentsline {subsection}{\numberline {76.3.1}\leavevmode {\color {Chapter }ReadPackage}}{1240}{subsection.76.3.1}
\contentsline {subsection}{\numberline {76.3.2}\leavevmode {\color {Chapter }TestPackageAvailability}}{1240}{subsection.76.3.2}
\contentsline {subsection}{\numberline {76.3.3}\leavevmode {\color {Chapter }IsPackageLoaded}}{1241}{subsection.76.3.3}
\contentsline {subsection}{\numberline {76.3.4}\leavevmode {\color {Chapter }IsPackageMarkedForLoading}}{1241}{subsection.76.3.4}
\contentsline {subsection}{\numberline {76.3.5}\leavevmode {\color {Chapter }TestPackage}}{1241}{subsection.76.3.5}
\contentsline {subsection}{\numberline {76.3.6}\leavevmode {\color {Chapter }InstalledPackageVersion}}{1242}{subsection.76.3.6}
\contentsline {subsection}{\numberline {76.3.7}\leavevmode {\color {Chapter }DirectoriesPackageLibrary}}{1242}{subsection.76.3.7}
\contentsline {subsection}{\numberline {76.3.8}\leavevmode {\color {Chapter }DirectoriesPackagePrograms}}{1243}{subsection.76.3.8}
\contentsline {subsection}{\numberline {76.3.9}\leavevmode {\color {Chapter }CompareVersionNumbers}}{1243}{subsection.76.3.9}
\contentsline {subsection}{\numberline {76.3.10}\leavevmode {\color {Chapter }DeclareAutoreadableVariables}}{1243}{subsection.76.3.10}
\contentsline {subsection}{\numberline {76.3.11}\leavevmode {\color {Chapter }Kernel modules in \textsf {GAP} packages}}{1244}{subsection.76.3.11}
\contentsline {subsection}{\numberline {76.3.12}\leavevmode {\color {Chapter }LoadDynamicModule}}{1244}{subsection.76.3.12}
\contentsline {subsection}{\numberline {76.3.13}\leavevmode {\color {Chapter }The PackageInfo.g File}}{1244}{subsection.76.3.13}
\contentsline {subsection}{\numberline {76.3.14}\leavevmode {\color {Chapter }ValidatePackageInfo}}{1244}{subsection.76.3.14}
\contentsline {subsection}{\numberline {76.3.15}\leavevmode {\color {Chapter }ShowPackageVariables}}{1245}{subsection.76.3.15}
\contentsline {subsection}{\numberline {76.3.16}\leavevmode {\color {Chapter }BibEntry}}{1246}{subsection.76.3.16}
\contentsline {subsection}{\numberline {76.3.17}\leavevmode {\color {Chapter }Cite}}{1247}{subsection.76.3.17}
\contentsline {section}{\numberline {76.4}\leavevmode {\color {Chapter }Guidelines for Writing a GAP Package}}{1247}{section.76.4}
\contentsline {section}{\numberline {76.5}\leavevmode {\color {Chapter }Structure of a GAP Package}}{1248}{section.76.5}
\contentsline {section}{\numberline {76.6}\leavevmode {\color {Chapter }Writing Documentation and Tools Needed}}{1251}{section.76.6}
\contentsline {section}{\numberline {76.7}\leavevmode {\color {Chapter }An Example of a GAP Package}}{1252}{section.76.7}
\contentsline {section}{\numberline {76.8}\leavevmode {\color {Chapter }File Structure}}{1253}{section.76.8}
\contentsline {section}{\numberline {76.9}\leavevmode {\color {Chapter }Creating the PackageInfo.g File}}{1254}{section.76.9}
\contentsline {section}{\numberline {76.10}\leavevmode {\color {Chapter }Functions and Variables and Choices of Their Names}}{1254}{section.76.10}
\contentsline {section}{\numberline {76.11}\leavevmode {\color {Chapter }Package Dependencies (Requesting one \textsf {GAP} Package from within Another)}}{1257}{section.76.11}
\contentsline {section}{\numberline {76.12}\leavevmode {\color {Chapter }Declaration and Implementation Part of a Package}}{1258}{section.76.12}
\contentsline {section}{\numberline {76.13}\leavevmode {\color {Chapter }Autoreadable Variables}}{1259}{section.76.13}
\contentsline {section}{\numberline {76.14}\leavevmode {\color {Chapter }Standalone Programs in a \textsf {GAP} Package}}{1260}{section.76.14}
\contentsline {subsection}{\numberline {76.14.1}\leavevmode {\color {Chapter }Installation of \textsf {GAP} Package Binaries}}{1260}{subsection.76.14.1}
\contentsline {subsection}{\numberline {76.14.2}\leavevmode {\color {Chapter }Test for the Existence of GAP Package Binaries}}{1260}{subsection.76.14.2}
\contentsline {subsection}{\numberline {76.14.3}\leavevmode {\color {Chapter }Calling of and Communication with External Binaries}}{1261}{subsection.76.14.3}
\contentsline {section}{\numberline {76.15}\leavevmode {\color {Chapter }Having an InfoClass}}{1262}{section.76.15}
\contentsline {section}{\numberline {76.16}\leavevmode {\color {Chapter }The Banner}}{1262}{section.76.16}
\contentsline {section}{\numberline {76.17}\leavevmode {\color {Chapter }Version Numbers}}{1262}{section.76.17}
\contentsline {section}{\numberline {76.18}\leavevmode {\color {Chapter }Testing a \textsf {GAP} package}}{1263}{section.76.18}
\contentsline {subsection}{\numberline {76.18.1}\leavevmode {\color {Chapter }Tests files for a GAP package}}{1264}{subsection.76.18.1}
\contentsline {subsection}{\numberline {76.18.2}\leavevmode {\color {Chapter }Testing \textsf {GAP} package loading}}{1264}{subsection.76.18.2}
\contentsline {subsection}{\numberline {76.18.3}\leavevmode {\color {Chapter }LoadAllPackages}}{1266}{subsection.76.18.3}
\contentsline {subsection}{\numberline {76.18.4}\leavevmode {\color {Chapter }Testing a \textsf {GAP} package with the \textsf {GAP} standard test suite}}{1266}{subsection.76.18.4}
\contentsline {section}{\numberline {76.19}\leavevmode {\color {Chapter }Access to the \textsf {GAP} Development Version}}{1267}{section.76.19}
\contentsline {section}{\numberline {76.20}\leavevmode {\color {Chapter }Version control and continuous integration for \textsf {GAP} packages}}{1267}{section.76.20}
\contentsline {section}{\numberline {76.21}\leavevmode {\color {Chapter }Selecting a license for a \textsf {GAP} Package}}{1267}{section.76.21}
\contentsline {section}{\numberline {76.22}\leavevmode {\color {Chapter }Releasing a GAP Package}}{1268}{section.76.22}
\contentsline {section}{\numberline {76.23}\leavevmode {\color {Chapter }The homepage of a Package}}{1269}{section.76.23}
\contentsline {section}{\numberline {76.24}\leavevmode {\color {Chapter }Some things to keep in mind}}{1269}{section.76.24}
\contentsline {section}{\numberline {76.25}\leavevmode {\color {Chapter }Package release checklists}}{1270}{section.76.25}
\contentsline {subsection}{\numberline {76.25.1}\leavevmode {\color {Chapter }Checklist for releasing a new package}}{1270}{subsection.76.25.1}
\contentsline {subsection}{\numberline {76.25.2}\leavevmode {\color {Chapter }Checklist for upgrading the package for the next major release of \textsf {GAP}}}{1271}{subsection.76.25.2}
\contentsline {chapter}{\numberline {77}\leavevmode {\color {Chapter }Replaced and Removed Command Names}}{1272}{chapter.77}
\contentsline {section}{\numberline {77.1}\leavevmode {\color {Chapter }Group Actions {\textendash } Name Changes}}{1272}{section.77.1}
\contentsline {section}{\numberline {77.2}\leavevmode {\color {Chapter }Package Interface {\textendash } Obsolete Functions and Name Changes}}{1273}{section.77.2}
\contentsline {section}{\numberline {77.3}\leavevmode {\color {Chapter }Normal Forms of Integer Matrices {\textendash } Name Changes}}{1273}{section.77.3}
\contentsline {section}{\numberline {77.4}\leavevmode {\color {Chapter }Miscellaneous Name Changes or Removed Names}}{1273}{section.77.4}
\contentsline {subsection}{\numberline {77.4.1}\leavevmode {\color {Chapter }InfoObsolete}}{1274}{subsection.77.4.1}
\contentsline {section}{\numberline {77.5}\leavevmode {\color {Chapter }The former .gaprc file}}{1274}{section.77.5}
\contentsline {section}{\numberline {77.6}\leavevmode {\color {Chapter }Semigroup properties}}{1274}{section.77.6}
\contentsline {chapter}{\numberline {78}\leavevmode {\color {Chapter }Method Selection}}{1276}{chapter.78}
\contentsline {section}{\numberline {78.1}\leavevmode {\color {Chapter }Operations and Methods}}{1276}{section.78.1}
\contentsline {subsection}{\numberline {78.1.1}\leavevmode {\color {Chapter }IsOperation}}{1276}{subsection.78.1.1}
\contentsline {subsection}{\numberline {78.1.2}\leavevmode {\color {Chapter }TypeOfOperation}}{1277}{subsection.78.1.2}
\contentsline {subsection}{\numberline {78.1.3}\leavevmode {\color {Chapter }ShowDeclarationsOfOperation}}{1277}{subsection.78.1.3}
\contentsline {subsection}{\numberline {78.1.4}\leavevmode {\color {Chapter }NewOperation}}{1277}{subsection.78.1.4}
\contentsline {subsection}{\numberline {78.1.5}\leavevmode {\color {Chapter }DeclareOperation}}{1277}{subsection.78.1.5}
\contentsline {section}{\numberline {78.2}\leavevmode {\color {Chapter }Constructors}}{1278}{section.78.2}
\contentsline {subsection}{\numberline {78.2.1}\leavevmode {\color {Chapter }NewConstructor}}{1279}{subsection.78.2.1}
\contentsline {subsection}{\numberline {78.2.2}\leavevmode {\color {Chapter }DeclareConstructor}}{1280}{subsection.78.2.2}
\contentsline {section}{\numberline {78.3}\leavevmode {\color {Chapter }Method Installation}}{1280}{section.78.3}
\contentsline {subsection}{\numberline {78.3.1}\leavevmode {\color {Chapter }InstallMethod}}{1280}{subsection.78.3.1}
\contentsline {subsection}{\numberline {78.3.2}\leavevmode {\color {Chapter }InstallOtherMethod}}{1281}{subsection.78.3.2}
\contentsline {subsection}{\numberline {78.3.3}\leavevmode {\color {Chapter }InstallMethodWithRandomSource}}{1281}{subsection.78.3.3}
\contentsline {section}{\numberline {78.4}\leavevmode {\color {Chapter }Applicable Methods and Method Selection}}{1281}{section.78.4}
\contentsline {section}{\numberline {78.5}\leavevmode {\color {Chapter }Partial Methods}}{1282}{section.78.5}
\contentsline {subsection}{\numberline {78.5.1}\leavevmode {\color {Chapter }TryNextMethod}}{1282}{subsection.78.5.1}
\contentsline {section}{\numberline {78.6}\leavevmode {\color {Chapter }Redispatching}}{1283}{section.78.6}
\contentsline {subsection}{\numberline {78.6.1}\leavevmode {\color {Chapter }RedispatchOnCondition}}{1283}{subsection.78.6.1}
\contentsline {section}{\numberline {78.7}\leavevmode {\color {Chapter }Immediate Methods}}{1283}{section.78.7}
\contentsline {subsection}{\numberline {78.7.1}\leavevmode {\color {Chapter }InstallImmediateMethod}}{1283}{subsection.78.7.1}
\contentsline {section}{\numberline {78.8}\leavevmode {\color {Chapter }Logical Implications}}{1284}{section.78.8}
\contentsline {subsection}{\numberline {78.8.1}\leavevmode {\color {Chapter }InstallTrueMethod}}{1284}{subsection.78.8.1}
\contentsline {subsection}{\numberline {78.8.2}\leavevmode {\color {Chapter }SuspendMethodReordering}}{1285}{subsection.78.8.2}
\contentsline {section}{\numberline {78.9}\leavevmode {\color {Chapter }Operations and Mathematical Terms}}{1285}{section.78.9}
\contentsline {chapter}{\numberline {79}\leavevmode {\color {Chapter }Creating New Objects}}{1287}{chapter.79}
\contentsline {section}{\numberline {79.1}\leavevmode {\color {Chapter }Creating Objects}}{1287}{section.79.1}
\contentsline {subsection}{\numberline {79.1.1}\leavevmode {\color {Chapter }Objectify}}{1287}{subsection.79.1.1}
\contentsline {subsection}{\numberline {79.1.2}\leavevmode {\color {Chapter }ObjectifyWithAttributes}}{1287}{subsection.79.1.2}
\contentsline {section}{\numberline {79.2}\leavevmode {\color {Chapter }Component Objects}}{1288}{section.79.2}
\contentsline {subsection}{\numberline {79.2.1}\leavevmode {\color {Chapter }NamesOfComponents}}{1289}{subsection.79.2.1}
\contentsline {section}{\numberline {79.3}\leavevmode {\color {Chapter }Positional Objects}}{1289}{section.79.3}
\contentsline {section}{\numberline {79.4}\leavevmode {\color {Chapter }Implementing New List Objects}}{1291}{section.79.4}
\contentsline {section}{\numberline {79.5}\leavevmode {\color {Chapter }Example {\textendash } Constructing Enumerators}}{1292}{section.79.5}
\contentsline {section}{\numberline {79.6}\leavevmode {\color {Chapter }Example {\textendash } Constructing Iterators}}{1294}{section.79.6}
\contentsline {section}{\numberline {79.7}\leavevmode {\color {Chapter }Arithmetic Issues in the Implementation of New Kinds of Lists}}{1295}{section.79.7}
\contentsline {section}{\numberline {79.8}\leavevmode {\color {Chapter }External Representation}}{1296}{section.79.8}
\contentsline {subsection}{\numberline {79.8.1}\leavevmode {\color {Chapter }ExtRepOfObj}}{1297}{subsection.79.8.1}
\contentsline {section}{\numberline {79.9}\leavevmode {\color {Chapter }Mutability and Copying}}{1297}{section.79.9}
\contentsline {section}{\numberline {79.10}\leavevmode {\color {Chapter }Global Variables in the Library}}{1299}{section.79.10}
\contentsline {subsection}{\numberline {79.10.1}\leavevmode {\color {Chapter }DeclareGlobalVariable}}{1300}{subsection.79.10.1}
\contentsline {subsection}{\numberline {79.10.2}\leavevmode {\color {Chapter }InstallValue}}{1300}{subsection.79.10.2}
\contentsline {subsection}{\numberline {79.10.3}\leavevmode {\color {Chapter }FlushCaches}}{1300}{subsection.79.10.3}
\contentsline {subsection}{\numberline {79.10.4}\leavevmode {\color {Chapter }DeclareGlobalFunction}}{1300}{subsection.79.10.4}
\contentsline {subsection}{\numberline {79.10.5}\leavevmode {\color {Chapter }DeclareSynonym}}{1301}{subsection.79.10.5}
\contentsline {section}{\numberline {79.11}\leavevmode {\color {Chapter }Declaration and Implementation Part}}{1301}{section.79.11}
\contentsline {chapter}{\numberline {80}\leavevmode {\color {Chapter }Examples of Extending the System}}{1303}{chapter.80}
\contentsline {section}{\numberline {80.1}\leavevmode {\color {Chapter }Addition of a Method}}{1303}{section.80.1}
\contentsline {section}{\numberline {80.2}\leavevmode {\color {Chapter }Extending the Range of Definition of an Existing Operation}}{1305}{section.80.2}
\contentsline {section}{\numberline {80.3}\leavevmode {\color {Chapter }Enforcing Property Tests}}{1305}{section.80.3}
\contentsline {section}{\numberline {80.4}\leavevmode {\color {Chapter }Adding a new Operation}}{1306}{section.80.4}
\contentsline {section}{\numberline {80.5}\leavevmode {\color {Chapter }Adding a new Attribute}}{1306}{section.80.5}
\contentsline {section}{\numberline {80.6}\leavevmode {\color {Chapter }Adding a new Representation}}{1307}{section.80.6}
\contentsline {section}{\numberline {80.7}\leavevmode {\color {Chapter }Components versus Attributes}}{1309}{section.80.7}
\contentsline {section}{\numberline {80.8}\leavevmode {\color {Chapter }Adding new Concepts}}{1309}{section.80.8}
\contentsline {subsection}{\numberline {80.8.1}\leavevmode {\color {Chapter }Example: M-groups}}{1310}{subsection.80.8.1}
\contentsline {subsection}{\numberline {80.8.2}\leavevmode {\color {Chapter }Example: Groups with a word length}}{1311}{subsection.80.8.2}
\contentsline {subsection}{\numberline {80.8.3}\leavevmode {\color {Chapter }Example: Groups with a decomposition as semidirect product}}{1311}{subsection.80.8.3}
\contentsline {section}{\numberline {80.9}\leavevmode {\color {Chapter }Creating Own Arithmetic Objects}}{1312}{section.80.9}
\contentsline {subsection}{\numberline {80.9.1}\leavevmode {\color {Chapter }ArithmeticElementCreator}}{1312}{subsection.80.9.1}
\contentsline {subsection}{\numberline {80.9.2}\leavevmode {\color {Chapter }Example: ArithmeticElementCreator}}{1313}{subsection.80.9.2}
\contentsline {chapter}{\numberline {81}\leavevmode {\color {Chapter }An Example {\textendash } Residue Class Rings}}{1315}{chapter.81}
\contentsline {section}{\numberline {81.1}\leavevmode {\color {Chapter }A First Attempt to Implement Elements of Residue Class Rings}}{1315}{section.81.1}
\contentsline {section}{\numberline {81.2}\leavevmode {\color {Chapter }Why Proceed in a Different Way?}}{1316}{section.81.2}
\contentsline {section}{\numberline {81.3}\leavevmode {\color {Chapter }A Second Attempt to Implement Elements of Residue Class Rings}}{1317}{section.81.3}
\contentsline {section}{\numberline {81.4}\leavevmode {\color {Chapter }Compatibility of Residue Class Rings with Prime Fields}}{1328}{section.81.4}
\contentsline {section}{\numberline {81.5}\leavevmode {\color {Chapter }Further Improvements in Implementing Residue Class Rings}}{1334}{section.81.5}
\contentsline {chapter}{\numberline {82}\leavevmode {\color {Chapter }An Example {\textendash } Designing Arithmetic Operations}}{1336}{chapter.82}
\contentsline {section}{\numberline {82.1}\leavevmode {\color {Chapter }New Arithmetic Operations vs. New Objects}}{1336}{section.82.1}
\contentsline {section}{\numberline {82.2}\leavevmode {\color {Chapter }Designing new Multiplicative Objects}}{1337}{section.82.2}
\contentsline {chapter}{\numberline {83}\leavevmode {\color {Chapter }Library Files}}{1344}{chapter.83}
\contentsline {section}{\numberline {83.1}\leavevmode {\color {Chapter }File Types}}{1344}{section.83.1}
\contentsline {section}{\numberline {83.2}\leavevmode {\color {Chapter }Finding Implementations in the Library}}{1344}{section.83.2}
\contentsline {section}{\numberline {83.3}\leavevmode {\color {Chapter }Undocumented Variables}}{1345}{section.83.3}
\contentsline {chapter}{\numberline {84}\leavevmode {\color {Chapter }Interface to the GAP Help System}}{1347}{chapter.84}
\contentsline {section}{\numberline {84.1}\leavevmode {\color {Chapter }Installing and Removing a Help Book}}{1347}{section.84.1}
\contentsline {subsection}{\numberline {84.1.1}\leavevmode {\color {Chapter }HELP{\textunderscore }ADD{\textunderscore }BOOK}}{1347}{subsection.84.1.1}
\contentsline {subsection}{\numberline {84.1.2}\leavevmode {\color {Chapter }HELP{\textunderscore }REMOVE{\textunderscore }BOOK}}{1348}{subsection.84.1.2}
\contentsline {section}{\numberline {84.2}\leavevmode {\color {Chapter }The manual.six File}}{1348}{section.84.2}
\contentsline {section}{\numberline {84.3}\leavevmode {\color {Chapter }The Help Book Handler}}{1348}{section.84.3}
\contentsline {section}{\numberline {84.4}\leavevmode {\color {Chapter }Introducing new Viewer for the Online Help}}{1350}{section.84.4}
\contentsline {subsection}{\numberline {84.4.1}\leavevmode {\color {Chapter }HELP{\textunderscore }VIEWER{\textunderscore }INFO}}{1350}{subsection.84.4.1}
\contentsline {chapter}{\numberline {85}\leavevmode {\color {Chapter }Function-Operation-Attribute Triples}}{1351}{chapter.85}
\contentsline {section}{\numberline {85.1}\leavevmode {\color {Chapter }Key Dependent Operations}}{1351}{section.85.1}
\contentsline {subsection}{\numberline {85.1.1}\leavevmode {\color {Chapter }KeyDependentOperation}}{1351}{subsection.85.1.1}
\contentsline {section}{\numberline {85.2}\leavevmode {\color {Chapter }In Parent Attributes}}{1353}{section.85.2}
\contentsline {subsection}{\numberline {85.2.1}\leavevmode {\color {Chapter }InParentFOA}}{1353}{subsection.85.2.1}
\contentsline {section}{\numberline {85.3}\leavevmode {\color {Chapter }Operation Functions}}{1353}{section.85.3}
\contentsline {subsection}{\numberline {85.3.1}\leavevmode {\color {Chapter }OrbitsishOperation}}{1354}{subsection.85.3.1}
\contentsline {subsection}{\numberline {85.3.2}\leavevmode {\color {Chapter }OrbitishFO}}{1355}{subsection.85.3.2}
\contentsline {subsection}{\numberline {85.3.3}\leavevmode {\color {Chapter }Example: Orbit and OrbitOp}}{1355}{subsection.85.3.3}
\contentsline {chapter}{\numberline {86}\leavevmode {\color {Chapter }Weak Pointers}}{1357}{chapter.86}
\contentsline {section}{\numberline {86.1}\leavevmode {\color {Chapter }Weak Pointer Objects}}{1357}{section.86.1}
\contentsline {subsection}{\numberline {86.1.1}\leavevmode {\color {Chapter }WeakPointerObj}}{1357}{subsection.86.1.1}
\contentsline {section}{\numberline {86.2}\leavevmode {\color {Chapter }Low Level Access Functions for Weak Pointer Objects}}{1358}{section.86.2}
\contentsline {subsection}{\numberline {86.2.1}\leavevmode {\color {Chapter }SetElmWPObj}}{1358}{subsection.86.2.1}
\contentsline {section}{\numberline {86.3}\leavevmode {\color {Chapter }Accessing Weak Pointer Objects as Lists}}{1359}{section.86.3}
\contentsline {section}{\numberline {86.4}\leavevmode {\color {Chapter }Copying Weak Pointer Objects}}{1359}{section.86.4}
\contentsline {section}{\numberline {86.5}\leavevmode {\color {Chapter }The GASMAN Interface for Weak Pointer Objects}}{1360}{section.86.5}
\contentsline {chapter}{\numberline {87}\leavevmode {\color {Chapter }More about Stabilizer Chains}}{1361}{chapter.87}
\contentsline {section}{\numberline {87.1}\leavevmode {\color {Chapter }Generalized Conjugation Technique}}{1361}{section.87.1}
\contentsline {section}{\numberline {87.2}\leavevmode {\color {Chapter }The General Backtrack Algorithm with Ordered Partitions}}{1362}{section.87.2}
\contentsline {subsection}{\numberline {87.2.1}\leavevmode {\color {Chapter }Internal representation of ordered partitions}}{1362}{subsection.87.2.1}
\contentsline {subsection}{\numberline {87.2.2}\leavevmode {\color {Chapter }Functions for setting up an R-base}}{1363}{subsection.87.2.2}
\contentsline {subsection}{\numberline {87.2.3}\leavevmode {\color {Chapter }Refinement functions for the backtrack search}}{1366}{subsection.87.2.3}
\contentsline {subsection}{\numberline {87.2.4}\leavevmode {\color {Chapter }Functions for meeting ordered partitions}}{1367}{subsection.87.2.4}
\contentsline {section}{\numberline {87.3}\leavevmode {\color {Chapter }Stabilizer Chains for Automorphisms Acting on Enumerators}}{1368}{section.87.3}
\contentsline {subsection}{\numberline {87.3.1}\leavevmode {\color {Chapter }An operation domain for automorphisms}}{1369}{subsection.87.3.1}
\contentsline {subsection}{\numberline {87.3.2}\leavevmode {\color {Chapter }Enumerators for cosets of characteristic factors}}{1370}{subsection.87.3.2}
\contentsline {subsection}{\numberline {87.3.3}\leavevmode {\color {Chapter }Making automorphisms act on such enumerators}}{1371}{subsection.87.3.3}
\contentsline {chapter}{References}{1381}{chapter*.3}
\contentsline {chapter}{Index}{1382}{section*.4}
