\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}}{25}{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}}{38}{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}}{39}{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}}{42}{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}
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\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}}{43}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}\leavevmode {\color {Chapter }ColorPrompt}}{43}{subsection.3.6.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }The Programming Language}}{45}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Language Overview}}{45}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Lexical Structure}}{46}{section.4.2}
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\contentsline {section}{\numberline {4.7}\leavevmode {\color {Chapter }Expressions}}{49}{section.4.7}
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\contentsline {section}{\numberline {4.9}\leavevmode {\color {Chapter }More About Global Variables}}{51}{section.4.9}
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\contentsline {subsection}{\numberline {4.9.2}\leavevmode {\color {Chapter }MakeReadOnlyGlobal}}{52}{subsection.4.9.2}
\contentsline {subsection}{\numberline {4.9.3}\leavevmode {\color {Chapter }MakeReadWriteGlobal}}{52}{subsection.4.9.3}
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\contentsline {subsection}{\numberline {4.9.11}\leavevmode {\color {Chapter }TemporaryGlobalVarName}}{54}{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}}{56}{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}
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\contentsline {section}{\numberline {4.16}\leavevmode {\color {Chapter }Procedure Calls}}{60}{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}}{63}{section.4.20}
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\contentsline {section}{\numberline {4.22}\leavevmode {\color {Chapter }Continue}}{65}{section.4.22}
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\contentsline {section}{\numberline {4.24}\leavevmode {\color {Chapter }Return (With or without Value)}}{68}{section.4.24}
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\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }NumberArgumentsFunction}}{70}{subsection.5.1.2}
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\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }FilenameFunc}}{71}{subsection.5.1.4}
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\contentsline {subsection}{\numberline {5.3.4}\leavevmode {\color {Chapter }IdFunc}}{74}{subsection.5.3.4}
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\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }View and Print}}{79}{section.6.3}
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\contentsline {section}{\numberline {6.6}\leavevmode {\color {Chapter }Error and ErrorCount}}{89}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}\leavevmode {\color {Chapter }Error}}{89}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}\leavevmode {\color {Chapter }ErrorCount}}{89}{subsection.6.6.2}
\contentsline {section}{\numberline {6.7}\leavevmode {\color {Chapter }Leaving GAP}}{90}{section.6.7}
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\contentsline {subsection}{\numberline {6.7.2}\leavevmode {\color {Chapter }InstallAtExit}}{90}{subsection.6.7.2}
\contentsline {subsection}{\numberline {6.7.3}\leavevmode {\color {Chapter }SaveOnExitFile}}{90}{subsection.6.7.3}
\contentsline {section}{\numberline {6.8}\leavevmode {\color {Chapter }Line Editing}}{90}{section.6.8}
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\contentsline {subsection}{\numberline {6.9.2}\leavevmode {\color {Chapter }The command line history}}{94}{subsection.6.9.2}
\contentsline {subsection}{\numberline {6.9.3}\leavevmode {\color {Chapter }SaveCommandLineHistory}}{94}{subsection.6.9.3}
\contentsline {subsection}{\numberline {6.9.4}\leavevmode {\color {Chapter }Writing your own command line editing functions}}{95}{subsection.6.9.4}
\contentsline {section}{\numberline {6.10}\leavevmode {\color {Chapter }Editing Files}}{96}{section.6.10}
\contentsline {subsection}{\numberline {6.10.1}\leavevmode {\color {Chapter }Edit}}{96}{subsection.6.10.1}
\contentsline {section}{\numberline {6.11}\leavevmode {\color {Chapter }Editor Support}}{96}{section.6.11}
\contentsline {section}{\numberline {6.12}\leavevmode {\color {Chapter }Changing the Screen Size}}{97}{section.6.12}
\contentsline {subsection}{\numberline {6.12.1}\leavevmode {\color {Chapter }SizeScreen}}{97}{subsection.6.12.1}
\contentsline {section}{\numberline {6.13}\leavevmode {\color {Chapter }Teaching Mode}}{97}{section.6.13}
\contentsline {subsection}{\numberline {6.13.1}\leavevmode {\color {Chapter }TeachingMode}}{98}{subsection.6.13.1}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Debugging and Profiling Facilities}}{99}{chapter.7}
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\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }Tracing Methods}}{101}{section.7.3}
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\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }UntraceMethods (for operations)}}{102}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }TraceImmediateMethods}}{102}{subsection.7.3.3}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }Info Functions}}{103}{section.7.4}
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\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Files and Filenames}}{124}{chapter.9}
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\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }GAP Root Directories}}{124}{section.9.2}
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\contentsline {section}{\numberline {39.7}\leavevmode {\color {Chapter }Cosets}}{502}{section.39.7}
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\contentsline {subsection}{\numberline {39.7.2}\leavevmode {\color {Chapter }RightCosets}}{503}{subsection.39.7.2}
\contentsline {subsection}{\numberline {39.7.3}\leavevmode {\color {Chapter }CanonicalRightCosetElement}}{503}{subsection.39.7.3}
\contentsline {subsection}{\numberline {39.7.4}\leavevmode {\color {Chapter }IsRightCoset}}{503}{subsection.39.7.4}
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\contentsline {section}{\numberline {39.8}\leavevmode {\color {Chapter }Transversals}}{504}{section.39.8}
\contentsline {subsection}{\numberline {39.8.1}\leavevmode {\color {Chapter }RightTransversal}}{504}{subsection.39.8.1}
\contentsline {section}{\numberline {39.9}\leavevmode {\color {Chapter }Double Cosets}}{504}{section.39.9}
\contentsline {subsection}{\numberline {39.9.1}\leavevmode {\color {Chapter }DoubleCoset}}{504}{subsection.39.9.1}
\contentsline {subsection}{\numberline {39.9.2}\leavevmode {\color {Chapter }RepresentativesContainedRightCosets}}{505}{subsection.39.9.2}
\contentsline {subsection}{\numberline {39.9.3}\leavevmode {\color {Chapter }DoubleCosets}}{505}{subsection.39.9.3}
\contentsline {subsection}{\numberline {39.9.4}\leavevmode {\color {Chapter }IsDoubleCoset (operation)}}{505}{subsection.39.9.4}
\contentsline {subsection}{\numberline {39.9.5}\leavevmode {\color {Chapter }DoubleCosetRepsAndSizes}}{505}{subsection.39.9.5}
\contentsline {subsection}{\numberline {39.9.6}\leavevmode {\color {Chapter }InfoCoset}}{506}{subsection.39.9.6}
\contentsline {section}{\numberline {39.10}\leavevmode {\color {Chapter }Conjugacy Classes}}{506}{section.39.10}
\contentsline {subsection}{\numberline {39.10.1}\leavevmode {\color {Chapter }ConjugacyClass}}{506}{subsection.39.10.1}
\contentsline {subsection}{\numberline {39.10.2}\leavevmode {\color {Chapter }ConjugacyClasses (attribute)}}{506}{subsection.39.10.2}
\contentsline {subsection}{\numberline {39.10.3}\leavevmode {\color {Chapter }ConjugacyClassesByRandomSearch}}{507}{subsection.39.10.3}
\contentsline {subsection}{\numberline {39.10.4}\leavevmode {\color {Chapter }ConjugacyClassesByOrbits}}{507}{subsection.39.10.4}
\contentsline {subsection}{\numberline {39.10.5}\leavevmode {\color {Chapter }NrConjugacyClasses}}{508}{subsection.39.10.5}
\contentsline {subsection}{\numberline {39.10.6}\leavevmode {\color {Chapter }RationalClass}}{508}{subsection.39.10.6}
\contentsline {subsection}{\numberline {39.10.7}\leavevmode {\color {Chapter }RationalClasses}}{508}{subsection.39.10.7}
\contentsline {subsection}{\numberline {39.10.8}\leavevmode {\color {Chapter }GaloisGroup (of rational class of a group)}}{508}{subsection.39.10.8}
\contentsline {subsection}{\numberline {39.10.9}\leavevmode {\color {Chapter }IsConjugate}}{509}{subsection.39.10.9}
\contentsline {subsection}{\numberline {39.10.10}\leavevmode {\color {Chapter }NthRootsInGroup}}{509}{subsection.39.10.10}
\contentsline {section}{\numberline {39.11}\leavevmode {\color {Chapter }Normal Structure}}{509}{section.39.11}
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\contentsline {subsection}{\numberline {39.11.2}\leavevmode {\color {Chapter }Core}}{510}{subsection.39.11.2}
\contentsline {subsection}{\numberline {39.11.3}\leavevmode {\color {Chapter }PCore}}{510}{subsection.39.11.3}
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\contentsline {subsection}{\numberline {39.11.6}\leavevmode {\color {Chapter }ComplementClassesRepresentatives}}{510}{subsection.39.11.6}
\contentsline {subsection}{\numberline {39.11.7}\leavevmode {\color {Chapter }InfoComplement}}{511}{subsection.39.11.7}
\contentsline {section}{\numberline {39.12}\leavevmode {\color {Chapter }Specific and Parametrized Subgroups}}{511}{section.39.12}
\contentsline {subsection}{\numberline {39.12.1}\leavevmode {\color {Chapter }TrivialSubgroup}}{511}{subsection.39.12.1}
\contentsline {subsection}{\numberline {39.12.2}\leavevmode {\color {Chapter }CommutatorSubgroup}}{511}{subsection.39.12.2}
\contentsline {subsection}{\numberline {39.12.3}\leavevmode {\color {Chapter }DerivedSubgroup}}{511}{subsection.39.12.3}
\contentsline {subsection}{\numberline {39.12.4}\leavevmode {\color {Chapter }CommutatorLength}}{512}{subsection.39.12.4}
\contentsline {subsection}{\numberline {39.12.5}\leavevmode {\color {Chapter }FittingSubgroup}}{512}{subsection.39.12.5}
\contentsline {subsection}{\numberline {39.12.6}\leavevmode {\color {Chapter }FrattiniSubgroup}}{512}{subsection.39.12.6}
\contentsline {subsection}{\numberline {39.12.7}\leavevmode {\color {Chapter }PrefrattiniSubgroup}}{512}{subsection.39.12.7}
\contentsline {subsection}{\numberline {39.12.8}\leavevmode {\color {Chapter }PerfectResiduum}}{513}{subsection.39.12.8}
\contentsline {subsection}{\numberline {39.12.9}\leavevmode {\color {Chapter }RadicalGroup}}{513}{subsection.39.12.9}
\contentsline {subsection}{\numberline {39.12.10}\leavevmode {\color {Chapter }Socle}}{513}{subsection.39.12.10}
\contentsline {subsection}{\numberline {39.12.11}\leavevmode {\color {Chapter }SupersolvableResiduum}}{513}{subsection.39.12.11}
\contentsline {subsection}{\numberline {39.12.12}\leavevmode {\color {Chapter }PRump}}{514}{subsection.39.12.12}
\contentsline {section}{\numberline {39.13}\leavevmode {\color {Chapter }Sylow Subgroups and Hall Subgroups}}{514}{section.39.13}
\contentsline {subsection}{\numberline {39.13.1}\leavevmode {\color {Chapter }SylowSubgroup}}{514}{subsection.39.13.1}
\contentsline {subsection}{\numberline {39.13.2}\leavevmode {\color {Chapter }SylowComplement}}{514}{subsection.39.13.2}
\contentsline {subsection}{\numberline {39.13.3}\leavevmode {\color {Chapter }HallSubgroup}}{515}{subsection.39.13.3}
\contentsline {subsection}{\numberline {39.13.4}\leavevmode {\color {Chapter }SylowSystem}}{515}{subsection.39.13.4}
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\contentsline {subsection}{\numberline {39.13.6}\leavevmode {\color {Chapter }HallSystem}}{516}{subsection.39.13.6}
\contentsline {section}{\numberline {39.14}\leavevmode {\color {Chapter }Subgroups characterized by prime powers}}{516}{section.39.14}
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\contentsline {subsection}{\numberline {39.14.2}\leavevmode {\color {Chapter }Agemo}}{516}{subsection.39.14.2}
\contentsline {section}{\numberline {39.15}\leavevmode {\color {Chapter }Group Properties}}{516}{section.39.15}
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\contentsline {subsection}{\numberline {39.15.2}\leavevmode {\color {Chapter }IsElementaryAbelian}}{517}{subsection.39.15.2}
\contentsline {subsection}{\numberline {39.15.3}\leavevmode {\color {Chapter }IsNilpotentGroup}}{517}{subsection.39.15.3}
\contentsline {subsection}{\numberline {39.15.4}\leavevmode {\color {Chapter }NilpotencyClassOfGroup}}{517}{subsection.39.15.4}
\contentsline {subsection}{\numberline {39.15.5}\leavevmode {\color {Chapter }IsPerfectGroup}}{517}{subsection.39.15.5}
\contentsline {subsection}{\numberline {39.15.6}\leavevmode {\color {Chapter }IsSolvableGroup}}{517}{subsection.39.15.6}
\contentsline {subsection}{\numberline {39.15.7}\leavevmode {\color {Chapter }IsPolycyclicGroup}}{518}{subsection.39.15.7}
\contentsline {subsection}{\numberline {39.15.8}\leavevmode {\color {Chapter }IsSupersolvableGroup}}{518}{subsection.39.15.8}
\contentsline {subsection}{\numberline {39.15.9}\leavevmode {\color {Chapter }IsMonomialGroup}}{518}{subsection.39.15.9}
\contentsline {subsection}{\numberline {39.15.10}\leavevmode {\color {Chapter }IsSimpleGroup}}{518}{subsection.39.15.10}
\contentsline {subsection}{\numberline {39.15.11}\leavevmode {\color {Chapter }IsAlmostSimpleGroup}}{518}{subsection.39.15.11}
\contentsline {subsection}{\numberline {39.15.12}\leavevmode {\color {Chapter }IsomorphismTypeInfoFiniteSimpleGroup}}{519}{subsection.39.15.12}
\contentsline {subsection}{\numberline {39.15.13}\leavevmode {\color {Chapter }SimpleGroup}}{520}{subsection.39.15.13}
\contentsline {subsection}{\numberline {39.15.14}\leavevmode {\color {Chapter }SimpleGroupsIterator}}{521}{subsection.39.15.14}
\contentsline {subsection}{\numberline {39.15.15}\leavevmode {\color {Chapter }SmallSimpleGroup}}{521}{subsection.39.15.15}
\contentsline {subsection}{\numberline {39.15.16}\leavevmode {\color {Chapter }AllSmallNonabelianSimpleGroups}}{521}{subsection.39.15.16}
\contentsline {subsection}{\numberline {39.15.17}\leavevmode {\color {Chapter }IsFinitelyGeneratedGroup}}{522}{subsection.39.15.17}
\contentsline {subsection}{\numberline {39.15.18}\leavevmode {\color {Chapter }IsSubsetLocallyFiniteGroup}}{522}{subsection.39.15.18}
\contentsline {subsection}{\numberline {39.15.19}\leavevmode {\color {Chapter }IsPGroup}}{522}{subsection.39.15.19}
\contentsline {subsection}{\numberline {39.15.20}\leavevmode {\color {Chapter }PrimePGroup}}{522}{subsection.39.15.20}
\contentsline {subsection}{\numberline {39.15.21}\leavevmode {\color {Chapter }PClassPGroup}}{523}{subsection.39.15.21}
\contentsline {subsection}{\numberline {39.15.22}\leavevmode {\color {Chapter }RankPGroup}}{523}{subsection.39.15.22}
\contentsline {subsection}{\numberline {39.15.23}\leavevmode {\color {Chapter }IsPSolvable}}{523}{subsection.39.15.23}
\contentsline {subsection}{\numberline {39.15.24}\leavevmode {\color {Chapter }IsPNilpotent}}{523}{subsection.39.15.24}
\contentsline {section}{\numberline {39.16}\leavevmode {\color {Chapter }Numerical Group Attributes}}{523}{section.39.16}
\contentsline {subsection}{\numberline {39.16.1}\leavevmode {\color {Chapter }AbelianInvariants}}{523}{subsection.39.16.1}
\contentsline {subsection}{\numberline {39.16.2}\leavevmode {\color {Chapter }Exponent}}{524}{subsection.39.16.2}
\contentsline {subsection}{\numberline {39.16.3}\leavevmode {\color {Chapter }EulerianFunction}}{524}{subsection.39.16.3}
\contentsline {section}{\numberline {39.17}\leavevmode {\color {Chapter }Subgroup Series}}{524}{section.39.17}
\contentsline {subsection}{\numberline {39.17.1}\leavevmode {\color {Chapter }ChiefSeries}}{525}{subsection.39.17.1}
\contentsline {subsection}{\numberline {39.17.2}\leavevmode {\color {Chapter }ChiefSeriesThrough}}{525}{subsection.39.17.2}
\contentsline {subsection}{\numberline {39.17.3}\leavevmode {\color {Chapter }ChiefSeriesUnderAction}}{525}{subsection.39.17.3}
\contentsline {subsection}{\numberline {39.17.4}\leavevmode {\color {Chapter }SubnormalSeries}}{525}{subsection.39.17.4}
\contentsline {subsection}{\numberline {39.17.5}\leavevmode {\color {Chapter }CompositionSeries}}{525}{subsection.39.17.5}
\contentsline {subsection}{\numberline {39.17.6}\leavevmode {\color {Chapter }DisplayCompositionSeries}}{526}{subsection.39.17.6}
\contentsline {subsection}{\numberline {39.17.7}\leavevmode {\color {Chapter }DerivedSeriesOfGroup}}{526}{subsection.39.17.7}
\contentsline {subsection}{\numberline {39.17.8}\leavevmode {\color {Chapter }DerivedLength}}{526}{subsection.39.17.8}
\contentsline {subsection}{\numberline {39.17.9}\leavevmode {\color {Chapter }ElementaryAbelianSeries}}{526}{subsection.39.17.9}
\contentsline {subsection}{\numberline {39.17.10}\leavevmode {\color {Chapter }InvariantElementaryAbelianSeries}}{527}{subsection.39.17.10}
\contentsline {subsection}{\numberline {39.17.11}\leavevmode {\color {Chapter }LowerCentralSeriesOfGroup}}{527}{subsection.39.17.11}
\contentsline {subsection}{\numberline {39.17.12}\leavevmode {\color {Chapter }UpperCentralSeriesOfGroup}}{527}{subsection.39.17.12}
\contentsline {subsection}{\numberline {39.17.13}\leavevmode {\color {Chapter }PCentralSeries}}{527}{subsection.39.17.13}
\contentsline {subsection}{\numberline {39.17.14}\leavevmode {\color {Chapter }JenningsSeries}}{528}{subsection.39.17.14}
\contentsline {subsection}{\numberline {39.17.15}\leavevmode {\color {Chapter }DimensionsLoewyFactors}}{528}{subsection.39.17.15}
\contentsline {subsection}{\numberline {39.17.16}\leavevmode {\color {Chapter }AscendingChain}}{528}{subsection.39.17.16}
\contentsline {subsection}{\numberline {39.17.17}\leavevmode {\color {Chapter }IntermediateGroup}}{528}{subsection.39.17.17}
\contentsline {subsection}{\numberline {39.17.18}\leavevmode {\color {Chapter }IntermediateSubgroups}}{528}{subsection.39.17.18}
\contentsline {section}{\numberline {39.18}\leavevmode {\color {Chapter }Factor Groups}}{529}{section.39.18}
\contentsline {subsection}{\numberline {39.18.1}\leavevmode {\color {Chapter }NaturalHomomorphismByNormalSubgroup}}{529}{subsection.39.18.1}
\contentsline {subsection}{\numberline {39.18.2}\leavevmode {\color {Chapter }FactorGroup}}{529}{subsection.39.18.2}
\contentsline {subsection}{\numberline {39.18.3}\leavevmode {\color {Chapter }CommutatorFactorGroup}}{529}{subsection.39.18.3}
\contentsline {subsection}{\numberline {39.18.4}\leavevmode {\color {Chapter }MaximalAbelianQuotient}}{530}{subsection.39.18.4}
\contentsline {subsection}{\numberline {39.18.5}\leavevmode {\color {Chapter }HasAbelianFactorGroup}}{530}{subsection.39.18.5}
\contentsline {subsection}{\numberline {39.18.6}\leavevmode {\color {Chapter }HasElementaryAbelianFactorGroup}}{530}{subsection.39.18.6}
\contentsline {subsection}{\numberline {39.18.7}\leavevmode {\color {Chapter }CentralizerModulo}}{530}{subsection.39.18.7}
\contentsline {section}{\numberline {39.19}\leavevmode {\color {Chapter }Sets of Subgroups}}{530}{section.39.19}
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\contentsline {subsection}{\numberline {39.19.2}\leavevmode {\color {Chapter }IsConjugacyClassSubgroupsRep}}{531}{subsection.39.19.2}
\contentsline {subsection}{\numberline {39.19.3}\leavevmode {\color {Chapter }ConjugacyClassesSubgroups}}{531}{subsection.39.19.3}
\contentsline {subsection}{\numberline {39.19.4}\leavevmode {\color {Chapter }ConjugacyClassesMaximalSubgroups}}{531}{subsection.39.19.4}
\contentsline {subsection}{\numberline {39.19.5}\leavevmode {\color {Chapter }AllSubgroups}}{532}{subsection.39.19.5}
\contentsline {subsection}{\numberline {39.19.6}\leavevmode {\color {Chapter }MaximalSubgroupClassReps}}{532}{subsection.39.19.6}
\contentsline {subsection}{\numberline {39.19.7}\leavevmode {\color {Chapter }MaximalSubgroups}}{532}{subsection.39.19.7}
\contentsline {subsection}{\numberline {39.19.8}\leavevmode {\color {Chapter }NormalSubgroups}}{532}{subsection.39.19.8}
\contentsline {subsection}{\numberline {39.19.9}\leavevmode {\color {Chapter }MaximalNormalSubgroups}}{533}{subsection.39.19.9}
\contentsline {subsection}{\numberline {39.19.10}\leavevmode {\color {Chapter }MinimalNormalSubgroups}}{533}{subsection.39.19.10}
\contentsline {section}{\numberline {39.20}\leavevmode {\color {Chapter }Subgroup Lattice}}{533}{section.39.20}
\contentsline {subsection}{\numberline {39.20.1}\leavevmode {\color {Chapter }LatticeSubgroups}}{533}{subsection.39.20.1}
\contentsline {subsection}{\numberline {39.20.2}\leavevmode {\color {Chapter }ClassElementLattice}}{533}{subsection.39.20.2}
\contentsline {subsection}{\numberline {39.20.3}\leavevmode {\color {Chapter }DotFileLatticeSubgroups}}{534}{subsection.39.20.3}
\contentsline {subsection}{\numberline {39.20.4}\leavevmode {\color {Chapter }MaximalSubgroupsLattice}}{534}{subsection.39.20.4}
\contentsline {subsection}{\numberline {39.20.5}\leavevmode {\color {Chapter }MinimalSupergroupsLattice}}{535}{subsection.39.20.5}
\contentsline {subsection}{\numberline {39.20.6}\leavevmode {\color {Chapter }RepresentativesPerfectSubgroups}}{535}{subsection.39.20.6}
\contentsline {subsection}{\numberline {39.20.7}\leavevmode {\color {Chapter }ConjugacyClassesPerfectSubgroups}}{536}{subsection.39.20.7}
\contentsline {subsection}{\numberline {39.20.8}\leavevmode {\color {Chapter }Zuppos}}{536}{subsection.39.20.8}
\contentsline {subsection}{\numberline {39.20.9}\leavevmode {\color {Chapter }InfoLattice}}{536}{subsection.39.20.9}
\contentsline {section}{\numberline {39.21}\leavevmode {\color {Chapter }Specific Methods for Subgroup Lattice Computations}}{536}{section.39.21}
\contentsline {subsection}{\numberline {39.21.1}\leavevmode {\color {Chapter }LatticeByCyclicExtension}}{536}{subsection.39.21.1}
\contentsline {subsection}{\numberline {39.21.2}\leavevmode {\color {Chapter }InvariantSubgroupsElementaryAbelianGroup}}{537}{subsection.39.21.2}
\contentsline {subsection}{\numberline {39.21.3}\leavevmode {\color {Chapter }SubgroupsSolvableGroup}}{537}{subsection.39.21.3}
\contentsline {subsection}{\numberline {39.21.4}\leavevmode {\color {Chapter }SizeConsiderFunction}}{539}{subsection.39.21.4}
\contentsline {subsection}{\numberline {39.21.5}\leavevmode {\color {Chapter }ExactSizeConsiderFunction}}{539}{subsection.39.21.5}
\contentsline {subsection}{\numberline {39.21.6}\leavevmode {\color {Chapter }InfoPcSubgroup}}{539}{subsection.39.21.6}
\contentsline {section}{\numberline {39.22}\leavevmode {\color {Chapter }Special Generating Sets}}{539}{section.39.22}
\contentsline {subsection}{\numberline {39.22.1}\leavevmode {\color {Chapter }GeneratorsSmallest}}{539}{subsection.39.22.1}
\contentsline {subsection}{\numberline {39.22.2}\leavevmode {\color {Chapter }LargestElementGroup}}{540}{subsection.39.22.2}
\contentsline {subsection}{\numberline {39.22.3}\leavevmode {\color {Chapter }MinimalGeneratingSet}}{540}{subsection.39.22.3}
\contentsline {subsection}{\numberline {39.22.4}\leavevmode {\color {Chapter }SmallGeneratingSet}}{540}{subsection.39.22.4}
\contentsline {subsection}{\numberline {39.22.5}\leavevmode {\color {Chapter }IndependentGeneratorsOfAbelianGroup}}{540}{subsection.39.22.5}
\contentsline {subsection}{\numberline {39.22.6}\leavevmode {\color {Chapter }IndependentGeneratorExponents}}{541}{subsection.39.22.6}
\contentsline {section}{\numberline {39.23}\leavevmode {\color {Chapter }1-Cohomology}}{541}{section.39.23}
\contentsline {subsection}{\numberline {39.23.1}\leavevmode {\color {Chapter }OneCocycles}}{541}{subsection.39.23.1}
\contentsline {subsection}{\numberline {39.23.2}\leavevmode {\color {Chapter }OneCoboundaries}}{542}{subsection.39.23.2}
\contentsline {subsection}{\numberline {39.23.3}\leavevmode {\color {Chapter }OCOneCocycles}}{543}{subsection.39.23.3}
\contentsline {subsection}{\numberline {39.23.4}\leavevmode {\color {Chapter }ComplementClassesRepresentativesEA}}{543}{subsection.39.23.4}
\contentsline {subsection}{\numberline {39.23.5}\leavevmode {\color {Chapter }InfoCoh}}{544}{subsection.39.23.5}
\contentsline {section}{\numberline {39.24}\leavevmode {\color {Chapter }Schur Covers and Multipliers}}{544}{section.39.24}
\contentsline {subsection}{\numberline {39.24.1}\leavevmode {\color {Chapter }EpimorphismSchurCover}}{544}{subsection.39.24.1}
\contentsline {subsection}{\numberline {39.24.2}\leavevmode {\color {Chapter }SchurCover}}{544}{subsection.39.24.2}
\contentsline {subsection}{\numberline {39.24.3}\leavevmode {\color {Chapter }AbelianInvariantsMultiplier}}{544}{subsection.39.24.3}
\contentsline {subsection}{\numberline {39.24.4}\leavevmode {\color {Chapter }Epicentre}}{545}{subsection.39.24.4}
\contentsline {subsection}{\numberline {39.24.5}\leavevmode {\color {Chapter }NonabelianExteriorSquare}}{545}{subsection.39.24.5}
\contentsline {subsection}{\numberline {39.24.6}\leavevmode {\color {Chapter }EpimorphismNonabelianExteriorSquare}}{545}{subsection.39.24.6}
\contentsline {subsection}{\numberline {39.24.7}\leavevmode {\color {Chapter }IsCentralFactor}}{545}{subsection.39.24.7}
\contentsline {subsection}{\numberline {39.24.8}\leavevmode {\color {Chapter }Covering groups of symmetric groups}}{546}{subsection.39.24.8}
\contentsline {subsection}{\numberline {39.24.9}\leavevmode {\color {Chapter }BasicSpinRepresentationOfSymmetricGroup}}{546}{subsection.39.24.9}
\contentsline {subsection}{\numberline {39.24.10}\leavevmode {\color {Chapter }SchurCoverOfSymmetricGroup}}{546}{subsection.39.24.10}
\contentsline {subsection}{\numberline {39.24.11}\leavevmode {\color {Chapter }DoubleCoverOfAlternatingGroup}}{547}{subsection.39.24.11}
\contentsline {section}{\numberline {39.25}\leavevmode {\color {Chapter }Tests for the Availability of Methods}}{547}{section.39.25}
\contentsline {subsection}{\numberline {39.25.1}\leavevmode {\color {Chapter }CanEasilyTestMembership}}{547}{subsection.39.25.1}
\contentsline {subsection}{\numberline {39.25.2}\leavevmode {\color {Chapter }CanEasilyComputeWithIndependentGensAbelianGroup}}{547}{subsection.39.25.2}
\contentsline {subsection}{\numberline {39.25.3}\leavevmode {\color {Chapter }CanComputeSize}}{548}{subsection.39.25.3}
\contentsline {subsection}{\numberline {39.25.4}\leavevmode {\color {Chapter }CanComputeSizeAnySubgroup}}{548}{subsection.39.25.4}
\contentsline {subsection}{\numberline {39.25.5}\leavevmode {\color {Chapter }CanComputeIndex}}{548}{subsection.39.25.5}
\contentsline {subsection}{\numberline {39.25.6}\leavevmode {\color {Chapter }CanComputeIsSubset}}{548}{subsection.39.25.6}
\contentsline {subsection}{\numberline {39.25.7}\leavevmode {\color {Chapter }KnowsHowToDecompose}}{548}{subsection.39.25.7}
\contentsline {chapter}{\numberline {40}\leavevmode {\color {Chapter }Group Homomorphisms}}{549}{chapter.40}
\contentsline {section}{\numberline {40.1}\leavevmode {\color {Chapter }Creating Group Homomorphisms}}{549}{section.40.1}
\contentsline {subsection}{\numberline {40.1.1}\leavevmode {\color {Chapter }GroupHomomorphismByImages}}{549}{subsection.40.1.1}
\contentsline {subsection}{\numberline {40.1.2}\leavevmode {\color {Chapter }GroupHomomorphismByImagesNC}}{550}{subsection.40.1.2}
\contentsline {subsection}{\numberline {40.1.3}\leavevmode {\color {Chapter }GroupGeneralMappingByImages}}{550}{subsection.40.1.3}
\contentsline {subsection}{\numberline {40.1.4}\leavevmode {\color {Chapter }GroupHomomorphismByFunction}}{551}{subsection.40.1.4}
\contentsline {subsection}{\numberline {40.1.5}\leavevmode {\color {Chapter }AsGroupGeneralMappingByImages}}{551}{subsection.40.1.5}
\contentsline {section}{\numberline {40.2}\leavevmode {\color {Chapter }Operations for Group Homomorphisms}}{552}{section.40.2}
\contentsline {section}{\numberline {40.3}\leavevmode {\color {Chapter }Efficiency of Homomorphisms}}{553}{section.40.3}
\contentsline {subsection}{\numberline {40.3.1}\leavevmode {\color {Chapter }Mappings given on generators}}{553}{subsection.40.3.1}
\contentsline {subsection}{\numberline {40.3.2}\leavevmode {\color {Chapter }Action homomorphisms}}{553}{subsection.40.3.2}
\contentsline {subsection}{\numberline {40.3.3}\leavevmode {\color {Chapter }Mappings given by functions}}{554}{subsection.40.3.3}
\contentsline {subsection}{\numberline {40.3.4}\leavevmode {\color {Chapter }Other operations}}{554}{subsection.40.3.4}
\contentsline {subsection}{\numberline {40.3.5}\leavevmode {\color {Chapter }ImagesSmallestGenerators}}{554}{subsection.40.3.5}
\contentsline {section}{\numberline {40.4}\leavevmode {\color {Chapter }Homomorphism for very large groups}}{554}{section.40.4}
\contentsline {section}{\numberline {40.5}\leavevmode {\color {Chapter }Nice Monomorphisms}}{555}{section.40.5}
\contentsline {subsection}{\numberline {40.5.1}\leavevmode {\color {Chapter }IsHandledByNiceMonomorphism}}{555}{subsection.40.5.1}
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\contentsline {chapter}{\numberline {44}\leavevmode {\color {Chapter }Matrix Groups}}{613}{chapter.44}
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\contentsline {section}{\numberline {44.2}\leavevmode {\color {Chapter }Attributes and Properties for Matrix Groups}}{614}{section.44.2}
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\contentsline {subsection}{\numberline {44.2.2}\leavevmode {\color {Chapter }DefaultFieldOfMatrixGroup}}{614}{subsection.44.2.2}
\contentsline {subsection}{\numberline {44.2.3}\leavevmode {\color {Chapter }FieldOfMatrixGroup}}{614}{subsection.44.2.3}
\contentsline {subsection}{\numberline {44.2.4}\leavevmode {\color {Chapter }TransposedMatrixGroup}}{614}{subsection.44.2.4}
\contentsline {subsection}{\numberline {44.2.5}\leavevmode {\color {Chapter }IsFFEMatrixGroup}}{614}{subsection.44.2.5}
\contentsline {section}{\numberline {44.3}\leavevmode {\color {Chapter }Actions of Matrix Groups}}{615}{section.44.3}
\contentsline {subsection}{\numberline {44.3.1}\leavevmode {\color {Chapter }ProjectiveActionOnFullSpace}}{615}{subsection.44.3.1}
\contentsline {subsection}{\numberline {44.3.2}\leavevmode {\color {Chapter }ProjectiveActionHomomorphismMatrixGroup}}{615}{subsection.44.3.2}
\contentsline {subsection}{\numberline {44.3.3}\leavevmode {\color {Chapter }BlowUpIsomorphism}}{615}{subsection.44.3.3}
\contentsline {section}{\numberline {44.4}\leavevmode {\color {Chapter }GL and SL}}{615}{section.44.4}
\contentsline {subsection}{\numberline {44.4.1}\leavevmode {\color {Chapter }IsGeneralLinearGroup}}{616}{subsection.44.4.1}
\contentsline {subsection}{\numberline {44.4.2}\leavevmode {\color {Chapter }IsNaturalGL}}{616}{subsection.44.4.2}
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\contentsline {section}{\numberline {44.5}\leavevmode {\color {Chapter }Invariant Forms}}{617}{section.44.5}
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\contentsline {subsection}{\numberline {44.5.2}\leavevmode {\color {Chapter }IsFullSubgroupGLorSLRespectingBilinearForm}}{617}{subsection.44.5.2}
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\contentsline {subsection}{\numberline {44.6.2}\leavevmode {\color {Chapter }IsRationalMatrixGroup}}{619}{subsection.44.6.2}
\contentsline {subsection}{\numberline {44.6.3}\leavevmode {\color {Chapter }IsIntegerMatrixGroup}}{619}{subsection.44.6.3}
\contentsline {subsection}{\numberline {44.6.4}\leavevmode {\color {Chapter }IsNaturalGLnZ}}{619}{subsection.44.6.4}
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\contentsline {subsection}{\numberline {44.6.8}\leavevmode {\color {Chapter }CentralizerInGLnZ}}{620}{subsection.44.6.8}
\contentsline {subsection}{\numberline {44.6.9}\leavevmode {\color {Chapter }ZClassRepsQClass}}{620}{subsection.44.6.9}
\contentsline {subsection}{\numberline {44.6.10}\leavevmode {\color {Chapter }IsBravaisGroup}}{620}{subsection.44.6.10}
\contentsline {subsection}{\numberline {44.6.11}\leavevmode {\color {Chapter }BravaisGroup}}{620}{subsection.44.6.11}
\contentsline {subsection}{\numberline {44.6.12}\leavevmode {\color {Chapter }BravaisSubgroups}}{620}{subsection.44.6.12}
\contentsline {subsection}{\numberline {44.6.13}\leavevmode {\color {Chapter }BravaisSupergroups}}{621}{subsection.44.6.13}
\contentsline {subsection}{\numberline {44.6.14}\leavevmode {\color {Chapter }NormalizerInGLnZBravaisGroup}}{621}{subsection.44.6.14}
\contentsline {section}{\numberline {44.7}\leavevmode {\color {Chapter }Acting OnRight and OnLeft}}{621}{section.44.7}
\contentsline {subsection}{\numberline {44.7.1}\leavevmode {\color {Chapter }CrystGroupDefaultAction}}{621}{subsection.44.7.1}
\contentsline {subsection}{\numberline {44.7.2}\leavevmode {\color {Chapter }SetCrystGroupDefaultAction}}{621}{subsection.44.7.2}
\contentsline {chapter}{\numberline {45}\leavevmode {\color {Chapter }Polycyclic Groups}}{622}{chapter.45}
\contentsline {section}{\numberline {45.1}\leavevmode {\color {Chapter }Polycyclic Generating Systems}}{622}{section.45.1}
\contentsline {section}{\numberline {45.2}\leavevmode {\color {Chapter }Computing a Pcgs}}{623}{section.45.2}
\contentsline {subsection}{\numberline {45.2.1}\leavevmode {\color {Chapter }Pcgs}}{623}{subsection.45.2.1}
\contentsline {subsection}{\numberline {45.2.2}\leavevmode {\color {Chapter }IsPcgs}}{623}{subsection.45.2.2}
\contentsline {subsection}{\numberline {45.2.3}\leavevmode {\color {Chapter }CanEasilyComputePcgs}}{623}{subsection.45.2.3}
\contentsline {section}{\numberline {45.3}\leavevmode {\color {Chapter }Defining a Pcgs Yourself}}{624}{section.45.3}
\contentsline {subsection}{\numberline {45.3.1}\leavevmode {\color {Chapter }PcgsByPcSequence}}{624}{subsection.45.3.1}
\contentsline {section}{\numberline {45.4}\leavevmode {\color {Chapter }Elementary Operations for a Pcgs}}{624}{section.45.4}
\contentsline {subsection}{\numberline {45.4.1}\leavevmode {\color {Chapter }RelativeOrders}}{624}{subsection.45.4.1}
\contentsline {subsection}{\numberline {45.4.2}\leavevmode {\color {Chapter }IsFiniteOrdersPcgs}}{624}{subsection.45.4.2}
\contentsline {subsection}{\numberline {45.4.3}\leavevmode {\color {Chapter }IsPrimeOrdersPcgs}}{624}{subsection.45.4.3}
\contentsline {subsection}{\numberline {45.4.4}\leavevmode {\color {Chapter }PcSeries}}{625}{subsection.45.4.4}
\contentsline {subsection}{\numberline {45.4.5}\leavevmode {\color {Chapter }GroupOfPcgs}}{625}{subsection.45.4.5}
\contentsline {subsection}{\numberline {45.4.6}\leavevmode {\color {Chapter }OneOfPcgs}}{625}{subsection.45.4.6}
\contentsline {section}{\numberline {45.5}\leavevmode {\color {Chapter }Elementary Operations for a Pcgs and an Element}}{625}{section.45.5}
\contentsline {subsection}{\numberline {45.5.1}\leavevmode {\color {Chapter }RelativeOrderOfPcElement}}{625}{subsection.45.5.1}
\contentsline {subsection}{\numberline {45.5.2}\leavevmode {\color {Chapter }ExponentOfPcElement}}{625}{subsection.45.5.2}
\contentsline {subsection}{\numberline {45.5.3}\leavevmode {\color {Chapter }ExponentsOfPcElement}}{625}{subsection.45.5.3}
\contentsline {subsection}{\numberline {45.5.4}\leavevmode {\color {Chapter }DepthOfPcElement}}{626}{subsection.45.5.4}
\contentsline {subsection}{\numberline {45.5.5}\leavevmode {\color {Chapter }LeadingExponentOfPcElement}}{626}{subsection.45.5.5}
\contentsline {subsection}{\numberline {45.5.6}\leavevmode {\color {Chapter }PcElementByExponents}}{626}{subsection.45.5.6}
\contentsline {subsection}{\numberline {45.5.7}\leavevmode {\color {Chapter }LinearCombinationPcgs}}{626}{subsection.45.5.7}
\contentsline {subsection}{\numberline {45.5.8}\leavevmode {\color {Chapter }SiftedPcElement}}{626}{subsection.45.5.8}
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\contentsline {subsection}{\numberline {45.5.10}\leavevmode {\color {Chapter }ReducedPcElement}}{627}{subsection.45.5.10}
\contentsline {subsection}{\numberline {45.5.11}\leavevmode {\color {Chapter }CleanedTailPcElement}}{627}{subsection.45.5.11}
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\contentsline {section}{\numberline {45.6}\leavevmode {\color {Chapter }Exponents of Special Products}}{627}{section.45.6}
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\contentsline {subsection}{\numberline {45.6.2}\leavevmode {\color {Chapter }ExponentsOfRelativePower}}{628}{subsection.45.6.2}
\contentsline {subsection}{\numberline {45.6.3}\leavevmode {\color {Chapter }ExponentsOfConjugate}}{628}{subsection.45.6.3}
\contentsline {subsection}{\numberline {45.6.4}\leavevmode {\color {Chapter }ExponentsOfCommutator}}{628}{subsection.45.6.4}
\contentsline {section}{\numberline {45.7}\leavevmode {\color {Chapter }Subgroups of Polycyclic Groups - Induced Pcgs}}{628}{section.45.7}
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\contentsline {subsection}{\numberline {45.7.2}\leavevmode {\color {Chapter }InducedPcgsByPcSequence}}{629}{subsection.45.7.2}
\contentsline {subsection}{\numberline {45.7.3}\leavevmode {\color {Chapter }ParentPcgs}}{629}{subsection.45.7.3}
\contentsline {subsection}{\numberline {45.7.4}\leavevmode {\color {Chapter }InducedPcgs}}{629}{subsection.45.7.4}
\contentsline {subsection}{\numberline {45.7.5}\leavevmode {\color {Chapter }InducedPcgsByGenerators}}{630}{subsection.45.7.5}
\contentsline {subsection}{\numberline {45.7.6}\leavevmode {\color {Chapter }InducedPcgsByPcSequenceAndGenerators}}{630}{subsection.45.7.6}
\contentsline {subsection}{\numberline {45.7.7}\leavevmode {\color {Chapter }LeadCoeffsIGS}}{630}{subsection.45.7.7}
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\contentsline {section}{\numberline {45.8}\leavevmode {\color {Chapter }Subgroups of Polycyclic Groups {\textendash } Canonical Pcgs}}{630}{section.45.8}
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\contentsline {section}{\numberline {45.9}\leavevmode {\color {Chapter }Factor Groups of Polycyclic Groups {\textendash } Modulo Pcgs}}{631}{section.45.9}
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\contentsline {subsection}{\numberline {45.9.4}\leavevmode {\color {Chapter }DenominatorOfModuloPcgs}}{632}{subsection.45.9.4}
\contentsline {subsection}{\numberline {45.9.5}\leavevmode {\color {Chapter }\texttt {\char 92\relax }mod (for two pcgs)}}{632}{subsection.45.9.5}
\contentsline {subsection}{\numberline {45.9.6}\leavevmode {\color {Chapter }CorrespondingGeneratorsByModuloPcgs}}{633}{subsection.45.9.6}
\contentsline {subsection}{\numberline {45.9.7}\leavevmode {\color {Chapter }CanonicalPcgsByGeneratorsWithImages}}{633}{subsection.45.9.7}
\contentsline {section}{\numberline {45.10}\leavevmode {\color {Chapter }Factor Groups of Polycyclic Groups in their Own Representation}}{633}{section.45.10}
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\contentsline {subsection}{\numberline {45.10.2}\leavevmode {\color {Chapter }ProjectedInducedPcgs}}{633}{subsection.45.10.2}
\contentsline {subsection}{\numberline {45.10.3}\leavevmode {\color {Chapter }LiftedPcElement}}{634}{subsection.45.10.3}
\contentsline {subsection}{\numberline {45.10.4}\leavevmode {\color {Chapter }LiftedInducedPcgs}}{634}{subsection.45.10.4}
\contentsline {section}{\numberline {45.11}\leavevmode {\color {Chapter }Pcgs and Normal Series}}{634}{section.45.11}
\contentsline {subsection}{\numberline {45.11.1}\leavevmode {\color {Chapter }IsPcgsElementaryAbelianSeries}}{634}{subsection.45.11.1}
\contentsline {subsection}{\numberline {45.11.2}\leavevmode {\color {Chapter }PcgsElementaryAbelianSeries (for a group)}}{634}{subsection.45.11.2}
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\contentsline {subsection}{\numberline {45.11.4}\leavevmode {\color {Chapter }EANormalSeriesByPcgs}}{635}{subsection.45.11.4}
\contentsline {subsection}{\numberline {45.11.5}\leavevmode {\color {Chapter }IsPcgsCentralSeries}}{635}{subsection.45.11.5}
\contentsline {subsection}{\numberline {45.11.6}\leavevmode {\color {Chapter }PcgsCentralSeries}}{635}{subsection.45.11.6}
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\contentsline {subsection}{\numberline {45.11.9}\leavevmode {\color {Chapter }IsPcgsPCentralSeriesPGroup}}{636}{subsection.45.11.9}
\contentsline {subsection}{\numberline {45.11.10}\leavevmode {\color {Chapter }PcgsPCentralSeriesPGroup}}{636}{subsection.45.11.10}
\contentsline {subsection}{\numberline {45.11.11}\leavevmode {\color {Chapter }IndicesPCentralNormalStepsPGroup}}{636}{subsection.45.11.11}
\contentsline {subsection}{\numberline {45.11.12}\leavevmode {\color {Chapter }PCentralNormalSeriesByPcgsPGroup}}{636}{subsection.45.11.12}
\contentsline {subsection}{\numberline {45.11.13}\leavevmode {\color {Chapter }IsPcgsChiefSeries}}{637}{subsection.45.11.13}
\contentsline {subsection}{\numberline {45.11.14}\leavevmode {\color {Chapter }PcgsChiefSeries}}{637}{subsection.45.11.14}
\contentsline {subsection}{\numberline {45.11.15}\leavevmode {\color {Chapter }IndicesChiefNormalSteps}}{637}{subsection.45.11.15}
\contentsline {subsection}{\numberline {45.11.16}\leavevmode {\color {Chapter }ChiefNormalSeriesByPcgs}}{637}{subsection.45.11.16}
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\contentsline {section}{\numberline {45.12}\leavevmode {\color {Chapter }Sum and Intersection of Pcgs}}{638}{section.45.12}
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\contentsline {section}{\numberline {45.13}\leavevmode {\color {Chapter }Special Pcgs}}{639}{section.45.13}
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\contentsline {subsection}{\numberline {45.13.2}\leavevmode {\color {Chapter }SpecialPcgs}}{640}{subsection.45.13.2}
\contentsline {subsection}{\numberline {45.13.3}\leavevmode {\color {Chapter }LGWeights}}{640}{subsection.45.13.3}
\contentsline {subsection}{\numberline {45.13.4}\leavevmode {\color {Chapter }LGLayers}}{640}{subsection.45.13.4}
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\contentsline {subsection}{\numberline {45.13.8}\leavevmode {\color {Chapter }InducedPcgsWrtSpecialPcgs}}{641}{subsection.45.13.8}
\contentsline {section}{\numberline {45.14}\leavevmode {\color {Chapter }Action on Subfactors Defined by a Pcgs}}{641}{section.45.14}
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\contentsline {subsection}{\numberline {45.14.2}\leavevmode {\color {Chapter }LinearAction}}{641}{subsection.45.14.2}
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\contentsline {section}{\numberline {45.15}\leavevmode {\color {Chapter }Orbit Stabilizer Methods for Polycyclic Groups}}{643}{section.45.15}
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\contentsline {subsection}{\numberline {45.15.2}\leavevmode {\color {Chapter }Pcgs{\textunderscore }OrbitStabilizer}}{643}{subsection.45.15.2}
\contentsline {section}{\numberline {45.16}\leavevmode {\color {Chapter }Operations which have Special Methods for Groups with Pcgs}}{643}{section.45.16}
\contentsline {section}{\numberline {45.17}\leavevmode {\color {Chapter }Conjugacy Classes in Solvable Groups}}{643}{section.45.17}
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\contentsline {subsection}{\numberline {45.17.2}\leavevmode {\color {Chapter }CentralizerSizeLimitConsiderFunction}}{644}{subsection.45.17.2}
\contentsline {chapter}{\numberline {46}\leavevmode {\color {Chapter }Pc Groups}}{645}{chapter.46}
\contentsline {section}{\numberline {46.1}\leavevmode {\color {Chapter }The family pcgs}}{646}{section.46.1}
\contentsline {subsection}{\numberline {46.1.1}\leavevmode {\color {Chapter }FamilyPcgs}}{646}{subsection.46.1.1}
\contentsline {subsection}{\numberline {46.1.2}\leavevmode {\color {Chapter }IsFamilyPcgs}}{646}{subsection.46.1.2}
\contentsline {subsection}{\numberline {46.1.3}\leavevmode {\color {Chapter }InducedPcgsWrtFamilyPcgs}}{646}{subsection.46.1.3}
\contentsline {subsection}{\numberline {46.1.4}\leavevmode {\color {Chapter }IsParentPcgsFamilyPcgs}}{646}{subsection.46.1.4}
\contentsline {section}{\numberline {46.2}\leavevmode {\color {Chapter }Elements of pc groups}}{647}{section.46.2}
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\contentsline {subsection}{\numberline {46.2.2}\leavevmode {\color {Chapter }Arithmetic operations for elements of pc groups}}{647}{subsection.46.2.2}
\contentsline {section}{\numberline {46.3}\leavevmode {\color {Chapter }Pc groups versus fp groups}}{647}{section.46.3}
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\contentsline {subsection}{\numberline {46.3.2}\leavevmode {\color {Chapter }IsomorphismFpGroupByPcgs}}{648}{subsection.46.3.2}
\contentsline {section}{\numberline {46.4}\leavevmode {\color {Chapter }Constructing Pc Groups}}{648}{section.46.4}
\contentsline {subsection}{\numberline {46.4.1}\leavevmode {\color {Chapter }PcGroupFpGroup}}{648}{subsection.46.4.1}
\contentsline {subsection}{\numberline {46.4.2}\leavevmode {\color {Chapter }SingleCollector}}{649}{subsection.46.4.2}
\contentsline {subsection}{\numberline {46.4.3}\leavevmode {\color {Chapter }SetConjugate}}{649}{subsection.46.4.3}
\contentsline {subsection}{\numberline {46.4.4}\leavevmode {\color {Chapter }SetCommutator}}{649}{subsection.46.4.4}
\contentsline {subsection}{\numberline {46.4.5}\leavevmode {\color {Chapter }SetPower}}{649}{subsection.46.4.5}
\contentsline {subsection}{\numberline {46.4.6}\leavevmode {\color {Chapter }GroupByRws}}{650}{subsection.46.4.6}
\contentsline {subsection}{\numberline {46.4.7}\leavevmode {\color {Chapter }IsConfluent (for pc groups)}}{650}{subsection.46.4.7}
\contentsline {subsection}{\numberline {46.4.8}\leavevmode {\color {Chapter }IsomorphismRefinedPcGroup}}{650}{subsection.46.4.8}
\contentsline {subsection}{\numberline {46.4.9}\leavevmode {\color {Chapter }RefinedPcGroup}}{650}{subsection.46.4.9}
\contentsline {section}{\numberline {46.5}\leavevmode {\color {Chapter }Computing Pc Groups}}{651}{section.46.5}
\contentsline {subsection}{\numberline {46.5.1}\leavevmode {\color {Chapter }PcGroupWithPcgs}}{651}{subsection.46.5.1}
\contentsline {subsection}{\numberline {46.5.2}\leavevmode {\color {Chapter }IsomorphismPcGroup}}{651}{subsection.46.5.2}
\contentsline {subsection}{\numberline {46.5.3}\leavevmode {\color {Chapter }IsomorphismSpecialPcGroup}}{652}{subsection.46.5.3}
\contentsline {section}{\numberline {46.6}\leavevmode {\color {Chapter }Saving a Pc Group}}{652}{section.46.6}
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\contentsline {section}{\numberline {46.7}\leavevmode {\color {Chapter }Operations for Pc Groups}}{652}{section.46.7}
\contentsline {section}{\numberline {46.8}\leavevmode {\color {Chapter }$2$-Cohomology and Extensions}}{652}{section.46.8}
\contentsline {subsection}{\numberline {46.8.1}\leavevmode {\color {Chapter }TwoCoboundaries}}{653}{subsection.46.8.1}
\contentsline {subsection}{\numberline {46.8.2}\leavevmode {\color {Chapter }TwoCocycles}}{653}{subsection.46.8.2}
\contentsline {subsection}{\numberline {46.8.3}\leavevmode {\color {Chapter }TwoCohomology}}{653}{subsection.46.8.3}
\contentsline {subsection}{\numberline {46.8.4}\leavevmode {\color {Chapter }Extensions}}{654}{subsection.46.8.4}
\contentsline {subsection}{\numberline {46.8.5}\leavevmode {\color {Chapter }Extension}}{654}{subsection.46.8.5}
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\contentsline {subsection}{\numberline {46.8.9}\leavevmode {\color {Chapter }ExtensionRepresentatives}}{655}{subsection.46.8.9}
\contentsline {subsection}{\numberline {46.8.10}\leavevmode {\color {Chapter }SplitExtensions}}{656}{subsection.46.8.10}
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\contentsline {chapter}{\numberline {47}\leavevmode {\color {Chapter }Finitely Presented Groups}}{658}{chapter.47}
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\contentsline {subsection}{\numberline {47.14.6}\leavevmode {\color {Chapter }EpimorphismSolvableQuotient}}{680}{subsection.47.14.6}
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\contentsline {subsection}{\numberline {47.15.4}\leavevmode {\color {Chapter }AbelianInvariantsNormalClosureFpGroup}}{681}{subsection.47.15.4}
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\contentsline {chapter}{\numberline {48}\leavevmode {\color {Chapter }Presentations and Tietze Transformations}}{684}{chapter.48}
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\contentsline {section}{\numberline {50.6}\leavevmode {\color {Chapter }Transitive Permutation Groups}}{732}{section.50.6}
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\contentsline {chapter}{\numberline {52}\leavevmode {\color {Chapter }Finitely Presented Semigroups and Monoids}}{779}{chapter.52}
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\contentsline {subsection}{\numberline {71.14.5}\leavevmode {\color {Chapter }IrreducibleRepresentationsDixon}}{1098}{subsection.71.14.5}
\contentsline {section}{\numberline {71.15}\leavevmode {\color {Chapter }Representations Given by Modules}}{1098}{section.71.15}
\contentsline {subsection}{\numberline {71.15.1}\leavevmode {\color {Chapter }IrreducibleModules}}{1098}{subsection.71.15.1}
\contentsline {subsection}{\numberline {71.15.2}\leavevmode {\color {Chapter }AbsolutelyIrreducibleModules}}{1098}{subsection.71.15.2}
\contentsline {subsection}{\numberline {71.15.3}\leavevmode {\color {Chapter }RegularModule}}{1099}{subsection.71.15.3}
\contentsline {section}{\numberline {71.16}\leavevmode {\color {Chapter }The Dixon-Schneider Algorithm}}{1099}{section.71.16}
\contentsline {section}{\numberline {71.17}\leavevmode {\color {Chapter }Advanced Methods for Dixon-Schneider Calculations}}{1099}{section.71.17}
\contentsline {subsection}{\numberline {71.17.1}\leavevmode {\color {Chapter }DixonRecord}}{1100}{subsection.71.17.1}
\contentsline {subsection}{\numberline {71.17.2}\leavevmode {\color {Chapter }DixonInit}}{1100}{subsection.71.17.2}
\contentsline {subsection}{\numberline {71.17.3}\leavevmode {\color {Chapter }DixontinI}}{1100}{subsection.71.17.3}
\contentsline {subsection}{\numberline {71.17.4}\leavevmode {\color {Chapter }DixonSplit}}{1100}{subsection.71.17.4}
\contentsline {subsection}{\numberline {71.17.5}\leavevmode {\color {Chapter }BestSplittingMatrix}}{1100}{subsection.71.17.5}
\contentsline {subsection}{\numberline {71.17.6}\leavevmode {\color {Chapter }DxIncludeIrreducibles}}{1101}{subsection.71.17.6}
\contentsline {subsection}{\numberline {71.17.7}\leavevmode {\color {Chapter }SplitCharacters}}{1101}{subsection.71.17.7}
\contentsline {subsection}{\numberline {71.17.8}\leavevmode {\color {Chapter }IsDxLargeGroup}}{1101}{subsection.71.17.8}
\contentsline {section}{\numberline {71.18}\leavevmode {\color {Chapter }Components of a Dixon Record}}{1101}{section.71.18}
\contentsline {section}{\numberline {71.19}\leavevmode {\color {Chapter }An Example of Advanced Dixon-Schneider Calculations}}{1102}{section.71.19}
\contentsline {section}{\numberline {71.20}\leavevmode {\color {Chapter }Constructing Character Tables from Others}}{1104}{section.71.20}
\contentsline {subsection}{\numberline {71.20.1}\leavevmode {\color {Chapter }CharacterTableDirectProduct}}{1104}{subsection.71.20.1}
\contentsline {subsection}{\numberline {71.20.2}\leavevmode {\color {Chapter }FactorsOfDirectProduct}}{1105}{subsection.71.20.2}
\contentsline {subsection}{\numberline {71.20.3}\leavevmode {\color {Chapter }CharacterTableFactorGroup}}{1105}{subsection.71.20.3}
\contentsline {subsection}{\numberline {71.20.4}\leavevmode {\color {Chapter }CharacterTableIsoclinic}}{1106}{subsection.71.20.4}
\contentsline {subsection}{\numberline {71.20.5}\leavevmode {\color {Chapter }CharacterTableWreathSymmetric}}{1106}{subsection.71.20.5}
\contentsline {section}{\numberline {71.21}\leavevmode {\color {Chapter }Sorted Character Tables}}{1107}{section.71.21}
\contentsline {subsection}{\numberline {71.21.1}\leavevmode {\color {Chapter }CharacterTableWithSortedCharacters}}{1107}{subsection.71.21.1}
\contentsline {subsection}{\numberline {71.21.2}\leavevmode {\color {Chapter }SortedCharacters}}{1108}{subsection.71.21.2}
\contentsline {subsection}{\numberline {71.21.3}\leavevmode {\color {Chapter }CharacterTableWithSortedClasses}}{1108}{subsection.71.21.3}
\contentsline {subsection}{\numberline {71.21.4}\leavevmode {\color {Chapter }SortedCharacterTable (w.r.t. a normal subgroup)}}{1108}{subsection.71.21.4}
\contentsline {subsection}{\numberline {71.21.5}\leavevmode {\color {Chapter }ClassPermutation}}{1109}{subsection.71.21.5}
\contentsline {section}{\numberline {71.22}\leavevmode {\color {Chapter }Automorphisms and Equivalence of Character Tables}}{1110}{section.71.22}
\contentsline {subsection}{\numberline {71.22.1}\leavevmode {\color {Chapter }MatrixAutomorphisms}}{1110}{subsection.71.22.1}
\contentsline {subsection}{\numberline {71.22.2}\leavevmode {\color {Chapter }TableAutomorphisms}}{1110}{subsection.71.22.2}
\contentsline {subsection}{\numberline {71.22.3}\leavevmode {\color {Chapter }TransformingPermutations}}{1111}{subsection.71.22.3}
\contentsline {subsection}{\numberline {71.22.4}\leavevmode {\color {Chapter }TransformingPermutationsCharacterTables}}{1111}{subsection.71.22.4}
\contentsline {subsection}{\numberline {71.22.5}\leavevmode {\color {Chapter }FamiliesOfRows}}{1112}{subsection.71.22.5}
\contentsline {section}{\numberline {71.23}\leavevmode {\color {Chapter }Storing Normal Subgroup Information}}{1112}{section.71.23}
\contentsline {subsection}{\numberline {71.23.1}\leavevmode {\color {Chapter }NormalSubgroupClassesInfo}}{1112}{subsection.71.23.1}
\contentsline {subsection}{\numberline {71.23.2}\leavevmode {\color {Chapter }ClassPositionsOfNormalSubgroup}}{1113}{subsection.71.23.2}
\contentsline {subsection}{\numberline {71.23.3}\leavevmode {\color {Chapter }NormalSubgroupClasses}}{1113}{subsection.71.23.3}
\contentsline {subsection}{\numberline {71.23.4}\leavevmode {\color {Chapter }FactorGroupNormalSubgroupClasses}}{1113}{subsection.71.23.4}
\contentsline {chapter}{\numberline {72}\leavevmode {\color {Chapter }Class Functions}}{1115}{chapter.72}
\contentsline {section}{\numberline {72.1}\leavevmode {\color {Chapter }Why Class Functions?}}{1115}{section.72.1}
\contentsline {subsection}{\numberline {72.1.1}\leavevmode {\color {Chapter }IsClassFunction}}{1117}{subsection.72.1.1}
\contentsline {section}{\numberline {72.2}\leavevmode {\color {Chapter }Basic Operations for Class Functions}}{1118}{section.72.2}
\contentsline {subsection}{\numberline {72.2.1}\leavevmode {\color {Chapter }UnderlyingCharacterTable}}{1118}{subsection.72.2.1}
\contentsline {subsection}{\numberline {72.2.2}\leavevmode {\color {Chapter }ValuesOfClassFunction}}{1118}{subsection.72.2.2}
\contentsline {section}{\numberline {72.3}\leavevmode {\color {Chapter }Comparison of Class Functions}}{1119}{section.72.3}
\contentsline {section}{\numberline {72.4}\leavevmode {\color {Chapter }Arithmetic Operations for Class Functions}}{1119}{section.72.4}
\contentsline {subsection}{\numberline {72.4.1}\leavevmode {\color {Chapter }Characteristic (for a class function)}}{1122}{subsection.72.4.1}
\contentsline {subsection}{\numberline {72.4.2}\leavevmode {\color {Chapter }ComplexConjugate (for a class function)}}{1122}{subsection.72.4.2}
\contentsline {subsection}{\numberline {72.4.3}\leavevmode {\color {Chapter }Order (for a class function)}}{1123}{subsection.72.4.3}
\contentsline {section}{\numberline {72.5}\leavevmode {\color {Chapter }Printing Class Functions}}{1123}{section.72.5}
\contentsline {subsection}{\numberline {72.5.1}\leavevmode {\color {Chapter }ViewObj (for class functions)}}{1123}{subsection.72.5.1}
\contentsline {subsection}{\numberline {72.5.2}\leavevmode {\color {Chapter }PrintObj (for class functions)}}{1123}{subsection.72.5.2}
\contentsline {subsection}{\numberline {72.5.3}\leavevmode {\color {Chapter }Display (for class functions)}}{1123}{subsection.72.5.3}
\contentsline {section}{\numberline {72.6}\leavevmode {\color {Chapter }Creating Class Functions from Values Lists}}{1124}{section.72.6}
\contentsline {subsection}{\numberline {72.6.1}\leavevmode {\color {Chapter }ClassFunction (for a character table and a list)}}{1124}{subsection.72.6.1}
\contentsline {subsection}{\numberline {72.6.2}\leavevmode {\color {Chapter }VirtualCharacter (for a character table and a list)}}{1124}{subsection.72.6.2}
\contentsline {subsection}{\numberline {72.6.3}\leavevmode {\color {Chapter }Character (for a character table and a list)}}{1124}{subsection.72.6.3}
\contentsline {subsection}{\numberline {72.6.4}\leavevmode {\color {Chapter }ClassFunctionSameType}}{1125}{subsection.72.6.4}
\contentsline {section}{\numberline {72.7}\leavevmode {\color {Chapter }Creating Class Functions using Groups}}{1125}{section.72.7}
\contentsline {subsection}{\numberline {72.7.1}\leavevmode {\color {Chapter }TrivialCharacter}}{1125}{subsection.72.7.1}
\contentsline {subsection}{\numberline {72.7.2}\leavevmode {\color {Chapter }NaturalCharacter (for a group)}}{1125}{subsection.72.7.2}
\contentsline {subsection}{\numberline {72.7.3}\leavevmode {\color {Chapter }PermutationCharacter}}{1126}{subsection.72.7.3}
\contentsline {section}{\numberline {72.8}\leavevmode {\color {Chapter }Operations for Class Functions}}{1126}{section.72.8}
\contentsline {subsection}{\numberline {72.8.1}\leavevmode {\color {Chapter }IsCharacter}}{1127}{subsection.72.8.1}
\contentsline {subsection}{\numberline {72.8.2}\leavevmode {\color {Chapter }IsVirtualCharacter}}{1127}{subsection.72.8.2}
\contentsline {subsection}{\numberline {72.8.3}\leavevmode {\color {Chapter }IsIrreducibleCharacter}}{1127}{subsection.72.8.3}
\contentsline {subsection}{\numberline {72.8.4}\leavevmode {\color {Chapter }DegreeOfCharacter}}{1127}{subsection.72.8.4}
\contentsline {subsection}{\numberline {72.8.5}\leavevmode {\color {Chapter }ScalarProduct (for characters)}}{1128}{subsection.72.8.5}
\contentsline {subsection}{\numberline {72.8.6}\leavevmode {\color {Chapter }MatScalarProducts}}{1128}{subsection.72.8.6}
\contentsline {subsection}{\numberline {72.8.7}\leavevmode {\color {Chapter }Norm (for a class function)}}{1128}{subsection.72.8.7}
\contentsline {subsection}{\numberline {72.8.8}\leavevmode {\color {Chapter }ConstituentsOfCharacter}}{1129}{subsection.72.8.8}
\contentsline {subsection}{\numberline {72.8.9}\leavevmode {\color {Chapter }KernelOfCharacter}}{1129}{subsection.72.8.9}
\contentsline {subsection}{\numberline {72.8.10}\leavevmode {\color {Chapter }ClassPositionsOfKernel}}{1129}{subsection.72.8.10}
\contentsline {subsection}{\numberline {72.8.11}\leavevmode {\color {Chapter }CentreOfCharacter}}{1130}{subsection.72.8.11}
\contentsline {subsection}{\numberline {72.8.12}\leavevmode {\color {Chapter }ClassPositionsOfCentre (for a character)}}{1130}{subsection.72.8.12}
\contentsline {subsection}{\numberline {72.8.13}\leavevmode {\color {Chapter }InertiaSubgroup}}{1130}{subsection.72.8.13}
\contentsline {subsection}{\numberline {72.8.14}\leavevmode {\color {Chapter }CycleStructureClass}}{1131}{subsection.72.8.14}
\contentsline {subsection}{\numberline {72.8.15}\leavevmode {\color {Chapter }IsTransitive (for a character)}}{1131}{subsection.72.8.15}
\contentsline {subsection}{\numberline {72.8.16}\leavevmode {\color {Chapter }Transitivity (for a character)}}{1131}{subsection.72.8.16}
\contentsline {subsection}{\numberline {72.8.17}\leavevmode {\color {Chapter }CentralCharacter}}{1131}{subsection.72.8.17}
\contentsline {subsection}{\numberline {72.8.18}\leavevmode {\color {Chapter }DeterminantOfCharacter}}{1131}{subsection.72.8.18}
\contentsline {subsection}{\numberline {72.8.19}\leavevmode {\color {Chapter }EigenvaluesChar}}{1132}{subsection.72.8.19}
\contentsline {subsection}{\numberline {72.8.20}\leavevmode {\color {Chapter }Tensored}}{1132}{subsection.72.8.20}
\contentsline {section}{\numberline {72.9}\leavevmode {\color {Chapter }Restricted and Induced Class Functions}}{1133}{section.72.9}
\contentsline {subsection}{\numberline {72.9.1}\leavevmode {\color {Chapter }RestrictedClassFunction}}{1133}{subsection.72.9.1}
\contentsline {subsection}{\numberline {72.9.2}\leavevmode {\color {Chapter }RestrictedClassFunctions}}{1133}{subsection.72.9.2}
\contentsline {subsection}{\numberline {72.9.3}\leavevmode {\color {Chapter }InducedClassFunction}}{1134}{subsection.72.9.3}
\contentsline {subsection}{\numberline {72.9.4}\leavevmode {\color {Chapter }InducedClassFunctions}}{1134}{subsection.72.9.4}
\contentsline {subsection}{\numberline {72.9.5}\leavevmode {\color {Chapter }InducedClassFunctionsByFusionMap}}{1134}{subsection.72.9.5}
\contentsline {subsection}{\numberline {72.9.6}\leavevmode {\color {Chapter }InducedCyclic}}{1135}{subsection.72.9.6}
\contentsline {section}{\numberline {72.10}\leavevmode {\color {Chapter }Reducing Virtual Characters}}{1135}{section.72.10}
\contentsline {subsection}{\numberline {72.10.1}\leavevmode {\color {Chapter }ReducedClassFunctions}}{1135}{subsection.72.10.1}
\contentsline {subsection}{\numberline {72.10.2}\leavevmode {\color {Chapter }ReducedCharacters}}{1136}{subsection.72.10.2}
\contentsline {subsection}{\numberline {72.10.3}\leavevmode {\color {Chapter }IrreducibleDifferences}}{1136}{subsection.72.10.3}
\contentsline {subsection}{\numberline {72.10.4}\leavevmode {\color {Chapter }LLL}}{1136}{subsection.72.10.4}
\contentsline {subsection}{\numberline {72.10.5}\leavevmode {\color {Chapter }Extract}}{1137}{subsection.72.10.5}
\contentsline {subsection}{\numberline {72.10.6}\leavevmode {\color {Chapter }OrthogonalEmbeddingsSpecialDimension}}{1138}{subsection.72.10.6}
\contentsline {subsection}{\numberline {72.10.7}\leavevmode {\color {Chapter }Decreased}}{1140}{subsection.72.10.7}
\contentsline {subsection}{\numberline {72.10.8}\leavevmode {\color {Chapter }DnLattice}}{1141}{subsection.72.10.8}
\contentsline {subsection}{\numberline {72.10.9}\leavevmode {\color {Chapter }DnLatticeIterative}}{1142}{subsection.72.10.9}
\contentsline {section}{\numberline {72.11}\leavevmode {\color {Chapter }Symmetrizations of Class Functions}}{1142}{section.72.11}
\contentsline {subsection}{\numberline {72.11.1}\leavevmode {\color {Chapter }Symmetrizations}}{1142}{subsection.72.11.1}
\contentsline {subsection}{\numberline {72.11.2}\leavevmode {\color {Chapter }SymmetricParts}}{1143}{subsection.72.11.2}
\contentsline {subsection}{\numberline {72.11.3}\leavevmode {\color {Chapter }AntiSymmetricParts}}{1143}{subsection.72.11.3}
\contentsline {subsection}{\numberline {72.11.4}\leavevmode {\color {Chapter }OrthogonalComponents}}{1143}{subsection.72.11.4}
\contentsline {subsection}{\numberline {72.11.5}\leavevmode {\color {Chapter }SymplecticComponents}}{1144}{subsection.72.11.5}
\contentsline {section}{\numberline {72.12}\leavevmode {\color {Chapter }Molien Series}}{1145}{section.72.12}
\contentsline {subsection}{\numberline {72.12.1}\leavevmode {\color {Chapter }MolienSeries}}{1145}{subsection.72.12.1}
\contentsline {subsection}{\numberline {72.12.2}\leavevmode {\color {Chapter }MolienSeriesInfo}}{1145}{subsection.72.12.2}
\contentsline {subsection}{\numberline {72.12.3}\leavevmode {\color {Chapter }ValueMolienSeries}}{1146}{subsection.72.12.3}
\contentsline {subsection}{\numberline {72.12.4}\leavevmode {\color {Chapter }MolienSeriesWithGivenDenominator}}{1147}{subsection.72.12.4}
\contentsline {section}{\numberline {72.13}\leavevmode {\color {Chapter }Possible Permutation Characters}}{1147}{section.72.13}
\contentsline {subsection}{\numberline {72.13.1}\leavevmode {\color {Chapter }PermCharInfo}}{1148}{subsection.72.13.1}
\contentsline {subsection}{\numberline {72.13.2}\leavevmode {\color {Chapter }PermCharInfoRelative}}{1149}{subsection.72.13.2}
\contentsline {section}{\numberline {72.14}\leavevmode {\color {Chapter }Computing Possible Permutation Characters}}{1150}{section.72.14}
\contentsline {subsection}{\numberline {72.14.1}\leavevmode {\color {Chapter }PermChars}}{1150}{subsection.72.14.1}
\contentsline {subsection}{\numberline {72.14.2}\leavevmode {\color {Chapter }TestPerm1, ..., TestPerm5}}{1152}{subsection.72.14.2}
\contentsline {subsection}{\numberline {72.14.3}\leavevmode {\color {Chapter }PermBounds}}{1153}{subsection.72.14.3}
\contentsline {subsection}{\numberline {72.14.4}\leavevmode {\color {Chapter }PermComb}}{1153}{subsection.72.14.4}
\contentsline {subsection}{\numberline {72.14.5}\leavevmode {\color {Chapter }Inequalities}}{1154}{subsection.72.14.5}
\contentsline {section}{\numberline {72.15}\leavevmode {\color {Chapter }Operations for Brauer Characters}}{1155}{section.72.15}
\contentsline {subsection}{\numberline {72.15.1}\leavevmode {\color {Chapter }FrobeniusCharacterValue}}{1155}{subsection.72.15.1}
\contentsline {subsection}{\numberline {72.15.2}\leavevmode {\color {Chapter }BrauerCharacterValue}}{1155}{subsection.72.15.2}
\contentsline {subsection}{\numberline {72.15.3}\leavevmode {\color {Chapter }SizeOfFieldOfDefinition}}{1155}{subsection.72.15.3}
\contentsline {subsection}{\numberline {72.15.4}\leavevmode {\color {Chapter }RealizableBrauerCharacters}}{1156}{subsection.72.15.4}
\contentsline {section}{\numberline {72.16}\leavevmode {\color {Chapter }Domains Generated by Class Functions}}{1156}{section.72.16}
\contentsline {chapter}{\numberline {73}\leavevmode {\color {Chapter }Maps Concerning Character Tables}}{1157}{chapter.73}
\contentsline {section}{\numberline {73.1}\leavevmode {\color {Chapter }Power Maps}}{1157}{section.73.1}
\contentsline {subsection}{\numberline {73.1.1}\leavevmode {\color {Chapter }PowerMap}}{1158}{subsection.73.1.1}
\contentsline {subsection}{\numberline {73.1.2}\leavevmode {\color {Chapter }PossiblePowerMaps}}{1159}{subsection.73.1.2}
\contentsline {subsection}{\numberline {73.1.3}\leavevmode {\color {Chapter }ElementOrdersPowerMap}}{1160}{subsection.73.1.3}
\contentsline {subsection}{\numberline {73.1.4}\leavevmode {\color {Chapter }PowerMapByComposition}}{1161}{subsection.73.1.4}
\contentsline {section}{\numberline {73.2}\leavevmode {\color {Chapter }Orbits on Sets of Possible Power Maps}}{1161}{section.73.2}
\contentsline {subsection}{\numberline {73.2.1}\leavevmode {\color {Chapter }OrbitPowerMaps}}{1161}{subsection.73.2.1}
\contentsline {subsection}{\numberline {73.2.2}\leavevmode {\color {Chapter }RepresentativesPowerMaps}}{1161}{subsection.73.2.2}
\contentsline {section}{\numberline {73.3}\leavevmode {\color {Chapter }Class Fusions between Character Tables}}{1162}{section.73.3}
\contentsline {subsection}{\numberline {73.3.1}\leavevmode {\color {Chapter }FusionConjugacyClasses}}{1163}{subsection.73.3.1}
\contentsline {subsection}{\numberline {73.3.2}\leavevmode {\color {Chapter }ComputedClassFusions}}{1164}{subsection.73.3.2}
\contentsline {subsection}{\numberline {73.3.3}\leavevmode {\color {Chapter }GetFusionMap}}{1164}{subsection.73.3.3}
\contentsline {subsection}{\numberline {73.3.4}\leavevmode {\color {Chapter }StoreFusion}}{1165}{subsection.73.3.4}
\contentsline {subsection}{\numberline {73.3.5}\leavevmode {\color {Chapter }NamesOfFusionSources}}{1165}{subsection.73.3.5}
\contentsline {subsection}{\numberline {73.3.6}\leavevmode {\color {Chapter }PossibleClassFusions}}{1166}{subsection.73.3.6}
\contentsline {subsection}{\numberline {73.3.7}\leavevmode {\color {Chapter }ConsiderStructureConstants}}{1167}{subsection.73.3.7}
\contentsline {section}{\numberline {73.4}\leavevmode {\color {Chapter }Orbits on Sets of Possible Class Fusions}}{1168}{section.73.4}
\contentsline {subsection}{\numberline {73.4.1}\leavevmode {\color {Chapter }OrbitFusions}}{1168}{subsection.73.4.1}
\contentsline {subsection}{\numberline {73.4.2}\leavevmode {\color {Chapter }RepresentativesFusions}}{1168}{subsection.73.4.2}
\contentsline {section}{\numberline {73.5}\leavevmode {\color {Chapter }Parametrized Maps}}{1168}{section.73.5}
\contentsline {subsection}{\numberline {73.5.1}\leavevmode {\color {Chapter }CompositionMaps}}{1169}{subsection.73.5.1}
\contentsline {subsection}{\numberline {73.5.2}\leavevmode {\color {Chapter }InverseMap}}{1169}{subsection.73.5.2}
\contentsline {subsection}{\numberline {73.5.3}\leavevmode {\color {Chapter }ProjectionMap}}{1170}{subsection.73.5.3}
\contentsline {subsection}{\numberline {73.5.4}\leavevmode {\color {Chapter }Indirected}}{1170}{subsection.73.5.4}
\contentsline {subsection}{\numberline {73.5.5}\leavevmode {\color {Chapter }Parametrized}}{1170}{subsection.73.5.5}
\contentsline {subsection}{\numberline {73.5.6}\leavevmode {\color {Chapter }ContainedMaps}}{1171}{subsection.73.5.6}
\contentsline {subsection}{\numberline {73.5.7}\leavevmode {\color {Chapter }UpdateMap}}{1171}{subsection.73.5.7}
\contentsline {subsection}{\numberline {73.5.8}\leavevmode {\color {Chapter }MeetMaps}}{1172}{subsection.73.5.8}
\contentsline {subsection}{\numberline {73.5.9}\leavevmode {\color {Chapter }CommutativeDiagram}}{1172}{subsection.73.5.9}
\contentsline {subsection}{\numberline {73.5.10}\leavevmode {\color {Chapter }CheckFixedPoints}}{1173}{subsection.73.5.10}
\contentsline {subsection}{\numberline {73.5.11}\leavevmode {\color {Chapter }TransferDiagram}}{1173}{subsection.73.5.11}
\contentsline {subsection}{\numberline {73.5.12}\leavevmode {\color {Chapter }TestConsistencyMaps}}{1174}{subsection.73.5.12}
\contentsline {subsection}{\numberline {73.5.13}\leavevmode {\color {Chapter }Indeterminateness}}{1174}{subsection.73.5.13}
\contentsline {subsection}{\numberline {73.5.14}\leavevmode {\color {Chapter }PrintAmbiguity}}{1175}{subsection.73.5.14}
\contentsline {subsection}{\numberline {73.5.15}\leavevmode {\color {Chapter }ContainedSpecialVectors}}{1175}{subsection.73.5.15}
\contentsline {subsection}{\numberline {73.5.16}\leavevmode {\color {Chapter }CollapsedMat}}{1176}{subsection.73.5.16}
\contentsline {subsection}{\numberline {73.5.17}\leavevmode {\color {Chapter }ContainedDecomposables}}{1177}{subsection.73.5.17}
\contentsline {section}{\numberline {73.6}\leavevmode {\color {Chapter }Subroutines for the Construction of Power Maps}}{1177}{section.73.6}
\contentsline {subsection}{\numberline {73.6.1}\leavevmode {\color {Chapter }InitPowerMap}}{1178}{subsection.73.6.1}
\contentsline {subsection}{\numberline {73.6.2}\leavevmode {\color {Chapter }Congruences (for character tables)}}{1178}{subsection.73.6.2}
\contentsline {subsection}{\numberline {73.6.3}\leavevmode {\color {Chapter }ConsiderKernels}}{1179}{subsection.73.6.3}
\contentsline {subsection}{\numberline {73.6.4}\leavevmode {\color {Chapter }ConsiderSmallerPowerMaps}}{1179}{subsection.73.6.4}
\contentsline {subsection}{\numberline {73.6.5}\leavevmode {\color {Chapter }MinusCharacter}}{1179}{subsection.73.6.5}
\contentsline {subsection}{\numberline {73.6.6}\leavevmode {\color {Chapter }PowerMapsAllowedBySymmetrizations}}{1180}{subsection.73.6.6}
\contentsline {section}{\numberline {73.7}\leavevmode {\color {Chapter }Subroutines for the Construction of Class Fusions}}{1181}{section.73.7}
\contentsline {subsection}{\numberline {73.7.1}\leavevmode {\color {Chapter }InitFusion}}{1181}{subsection.73.7.1}
\contentsline {subsection}{\numberline {73.7.2}\leavevmode {\color {Chapter }CheckPermChar}}{1181}{subsection.73.7.2}
\contentsline {subsection}{\numberline {73.7.3}\leavevmode {\color {Chapter }ConsiderTableAutomorphisms}}{1182}{subsection.73.7.3}
\contentsline {subsection}{\numberline {73.7.4}\leavevmode {\color {Chapter }FusionsAllowedByRestrictions}}{1182}{subsection.73.7.4}
\contentsline {chapter}{\numberline {74}\leavevmode {\color {Chapter }Unknowns}}{1184}{chapter.74}
\contentsline {section}{\numberline {74.1}\leavevmode {\color {Chapter }More about Unknowns}}{1184}{section.74.1}
\contentsline {subsection}{\numberline {74.1.1}\leavevmode {\color {Chapter }Unknown}}{1184}{subsection.74.1.1}
\contentsline {subsection}{\numberline {74.1.2}\leavevmode {\color {Chapter }LargestUnknown}}{1185}{subsection.74.1.2}
\contentsline {subsection}{\numberline {74.1.3}\leavevmode {\color {Chapter }IsUnknown}}{1185}{subsection.74.1.3}
\contentsline {subsection}{\numberline {74.1.4}\leavevmode {\color {Chapter }Comparison of Unknowns}}{1185}{subsection.74.1.4}
\contentsline {subsection}{\numberline {74.1.5}\leavevmode {\color {Chapter }Arithmetical Operations for Unknowns}}{1185}{subsection.74.1.5}
\contentsline {chapter}{\numberline {75}\leavevmode {\color {Chapter }Monomiality Questions}}{1187}{chapter.75}
\contentsline {section}{\numberline {75.1}\leavevmode {\color {Chapter }InfoMonomial (Info Class)}}{1188}{section.75.1}
\contentsline {subsection}{\numberline {75.1.1}\leavevmode {\color {Chapter }InfoMonomial}}{1188}{subsection.75.1.1}
\contentsline {section}{\numberline {75.2}\leavevmode {\color {Chapter }Character Degrees and Derived Length}}{1188}{section.75.2}
\contentsline {subsection}{\numberline {75.2.1}\leavevmode {\color {Chapter }Alpha}}{1188}{subsection.75.2.1}
\contentsline {subsection}{\numberline {75.2.2}\leavevmode {\color {Chapter }Delta}}{1188}{subsection.75.2.2}
\contentsline {subsection}{\numberline {75.2.3}\leavevmode {\color {Chapter }IsBergerCondition}}{1188}{subsection.75.2.3}
\contentsline {section}{\numberline {75.3}\leavevmode {\color {Chapter }Primitivity of Characters}}{1189}{section.75.3}
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\contentsline {subsection}{\numberline {75.5.2}\leavevmode {\color {Chapter }MinimalNonmonomialGroup}}{1195}{subsection.75.5.2}
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\contentsline {chapter}{\numberline {79}\leavevmode {\color {Chapter }Creating New Objects}}{1216}{chapter.79}
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\contentsline {chapter}{\numberline {80}\leavevmode {\color {Chapter }Examples of Extending the System}}{1238}{chapter.80}
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\contentsline {subsection}{\numberline {80.8.2}\leavevmode {\color {Chapter }Example: Groups with a word length}}{1246}{subsection.80.8.2}
\contentsline {subsection}{\numberline {80.8.3}\leavevmode {\color {Chapter }Example: Groups with a decomposition as semidirect product}}{1246}{subsection.80.8.3}
\contentsline {section}{\numberline {80.9}\leavevmode {\color {Chapter }Creating Own Arithmetic Objects}}{1247}{section.80.9}
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\contentsline {chapter}{\numberline {81}\leavevmode {\color {Chapter }An Example {\textendash } Residue Class Rings}}{1250}{chapter.81}
\contentsline {section}{\numberline {81.1}\leavevmode {\color {Chapter }A First Attempt to Implement Elements of Residue Class Rings}}{1250}{section.81.1}
\contentsline {section}{\numberline {81.2}\leavevmode {\color {Chapter }Why Proceed in a Different Way?}}{1251}{section.81.2}
\contentsline {section}{\numberline {81.3}\leavevmode {\color {Chapter }A Second Attempt to Implement Elements of Residue Class Rings}}{1252}{section.81.3}
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\contentsline {chapter}{\numberline {82}\leavevmode {\color {Chapter }An Example {\textendash } Designing Arithmetic Operations}}{1271}{chapter.82}
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\contentsline {chapter}{\numberline {84}\leavevmode {\color {Chapter }Interface to the GAP Help System}}{1281}{chapter.84}
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\contentsline {section}{\numberline {84.3}\leavevmode {\color {Chapter }The Help Book Handler}}{1282}{section.84.3}
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\contentsline {chapter}{\numberline {86}\leavevmode {\color {Chapter }Weak Pointers}}{1291}{chapter.86}
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\contentsline {subsection}{\numberline {86.2.1}\leavevmode {\color {Chapter }SetElmWPObj}}{1292}{subsection.86.2.1}
\contentsline {section}{\numberline {86.3}\leavevmode {\color {Chapter }Accessing Weak Pointer Objects as Lists}}{1293}{section.86.3}
\contentsline {section}{\numberline {86.4}\leavevmode {\color {Chapter }Copying Weak Pointer Objects}}{1293}{section.86.4}
\contentsline {section}{\numberline {86.5}\leavevmode {\color {Chapter }The GASMAN Interface for Weak Pointer Objects}}{1294}{section.86.5}
\contentsline {chapter}{\numberline {87}\leavevmode {\color {Chapter }More about Stabilizer Chains}}{1295}{chapter.87}
\contentsline {section}{\numberline {87.1}\leavevmode {\color {Chapter }Generalized Conjugation Technique}}{1295}{section.87.1}
\contentsline {section}{\numberline {87.2}\leavevmode {\color {Chapter }The General Backtrack Algorithm with Ordered Partitions}}{1296}{section.87.2}
\contentsline {subsection}{\numberline {87.2.1}\leavevmode {\color {Chapter }Internal representation of ordered partitions}}{1296}{subsection.87.2.1}
\contentsline {subsection}{\numberline {87.2.2}\leavevmode {\color {Chapter }Functions for setting up an R-base}}{1297}{subsection.87.2.2}
\contentsline {subsection}{\numberline {87.2.3}\leavevmode {\color {Chapter }Refinement functions for the backtrack search}}{1300}{subsection.87.2.3}
\contentsline {subsection}{\numberline {87.2.4}\leavevmode {\color {Chapter }Functions for meeting ordered partitions}}{1302}{subsection.87.2.4}
\contentsline {subsection}{\numberline {87.2.5}\leavevmode {\color {Chapter }Avoiding multiplication of permutations}}{1303}{subsection.87.2.5}
\contentsline {section}{\numberline {87.3}\leavevmode {\color {Chapter }Stabilizer Chains for Automorphisms Acting on Enumerators}}{1304}{section.87.3}
\contentsline {subsection}{\numberline {87.3.1}\leavevmode {\color {Chapter }An operation domain for automorphisms}}{1304}{subsection.87.3.1}
\contentsline {subsection}{\numberline {87.3.2}\leavevmode {\color {Chapter }Enumerators for cosets of characteristic factors}}{1305}{subsection.87.3.2}
\contentsline {subsection}{\numberline {87.3.3}\leavevmode {\color {Chapter }Making automorphisms act on such enumerators}}{1306}{subsection.87.3.3}
\contentsline {chapter}{References}{1318}{chapter*.3}
\contentsline {chapter}{Index}{1319}{section*.4}
