\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{Preface}}{24}{chapter.1}%
\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{The \textsf {GAP} System}}{24}{section.1.1}%
\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{Authors and Maintainers}}{26}{section.1.2}%
\contentsline {section}{\numberline {1.3}\textcolor {Chapter }{Acknowledgements}}{27}{section.1.3}%
\contentsline {section}{\numberline {1.4}\textcolor {Chapter }{Copyright and License}}{27}{section.1.4}%
\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{The Help System}}{28}{chapter.2}%
\contentsline {section}{\numberline {2.1}\textcolor {Chapter }{Invoking the Help}}{28}{section.2.1}%
\contentsline {section}{\numberline {2.2}\textcolor {Chapter }{Browsing through the Sections}}{28}{section.2.2}%
\contentsline {section}{\numberline {2.3}\textcolor {Chapter }{Changing the Help Viewer}}{29}{section.2.3}%
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\contentsline {section}{\numberline {2.4}\textcolor {Chapter }{The Pager Command}}{31}{section.2.4}%
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\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{Running GAP}}{33}{chapter.3}%
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\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{The gap.ini and gaprc files}}{38}{section.3.2}%
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\contentsline {subsection}{\numberline {3.2.2}\textcolor {Chapter }{The gaprc file}}{39}{subsection.3.2.2}%
\contentsline {subsection}{\numberline {3.2.3}\textcolor {Chapter }{Configuring User preferences}}{39}{subsection.3.2.3}%
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\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{The Programming Language}}{49}{chapter.4}%
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\contentsline {section}{\numberline {6.12}\textcolor {Chapter }{Changing the Screen Size}}{107}{section.6.12}%
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\contentsline {subsection}{\numberline {24.7.2}\textcolor {Chapter }{TriangulizedMat}}{363}{subsection.24.7.2}%
\contentsline {subsection}{\numberline {24.7.3}\textcolor {Chapter }{TriangulizeMat}}{363}{subsection.24.7.3}%
\contentsline {subsection}{\numberline {24.7.4}\textcolor {Chapter }{NullspaceMat}}{363}{subsection.24.7.4}%
\contentsline {subsection}{\numberline {24.7.5}\textcolor {Chapter }{NullspaceMatDestructive}}{363}{subsection.24.7.5}%
\contentsline {subsection}{\numberline {24.7.6}\textcolor {Chapter }{SolutionMat}}{364}{subsection.24.7.6}%
\contentsline {subsection}{\numberline {24.7.7}\textcolor {Chapter }{SolutionMatDestructive}}{364}{subsection.24.7.7}%
\contentsline {subsection}{\numberline {24.7.8}\textcolor {Chapter }{BaseFixedSpace}}{364}{subsection.24.7.8}%
\contentsline {section}{\numberline {24.8}\textcolor {Chapter }{Eigenvectors and eigenvalues}}{365}{section.24.8}%
\contentsline {subsection}{\numberline {24.8.1}\textcolor {Chapter }{GeneralisedEigenvalues}}{365}{subsection.24.8.1}%
\contentsline {subsection}{\numberline {24.8.2}\textcolor {Chapter }{GeneralisedEigenspaces}}{365}{subsection.24.8.2}%
\contentsline {subsection}{\numberline {24.8.3}\textcolor {Chapter }{Eigenvalues}}{365}{subsection.24.8.3}%
\contentsline {subsection}{\numberline {24.8.4}\textcolor {Chapter }{Eigenspaces}}{365}{subsection.24.8.4}%
\contentsline {subsection}{\numberline {24.8.5}\textcolor {Chapter }{Eigenvectors}}{365}{subsection.24.8.5}%
\contentsline {section}{\numberline {24.9}\textcolor {Chapter }{Elementary Divisors}}{365}{section.24.9}%
\contentsline {subsection}{\numberline {24.9.1}\textcolor {Chapter }{ElementaryDivisorsMat}}{365}{subsection.24.9.1}%
\contentsline {subsection}{\numberline {24.9.2}\textcolor {Chapter }{ElementaryDivisorsTransformationsMat}}{366}{subsection.24.9.2}%
\contentsline {subsection}{\numberline {24.9.3}\textcolor {Chapter }{DiagonalizeMat}}{367}{subsection.24.9.3}%
\contentsline {section}{\numberline {24.10}\textcolor {Chapter }{Echelonized Matrices}}{367}{section.24.10}%
\contentsline {subsection}{\numberline {24.10.1}\textcolor {Chapter }{SemiEchelonMat}}{367}{subsection.24.10.1}%
\contentsline {subsection}{\numberline {24.10.2}\textcolor {Chapter }{SemiEchelonMatDestructive}}{368}{subsection.24.10.2}%
\contentsline {subsection}{\numberline {24.10.3}\textcolor {Chapter }{SemiEchelonMatTransformation}}{368}{subsection.24.10.3}%
\contentsline {subsection}{\numberline {24.10.4}\textcolor {Chapter }{SemiEchelonMats}}{368}{subsection.24.10.4}%
\contentsline {subsection}{\numberline {24.10.5}\textcolor {Chapter }{SemiEchelonMatsDestructive}}{369}{subsection.24.10.5}%
\contentsline {section}{\numberline {24.11}\textcolor {Chapter }{Matrices as Basis of a Row Space}}{369}{section.24.11}%
\contentsline {subsection}{\numberline {24.11.1}\textcolor {Chapter }{BaseMat}}{369}{subsection.24.11.1}%
\contentsline {subsection}{\numberline {24.11.2}\textcolor {Chapter }{BaseMatDestructive}}{369}{subsection.24.11.2}%
\contentsline {subsection}{\numberline {24.11.3}\textcolor {Chapter }{BaseOrthogonalSpaceMat}}{369}{subsection.24.11.3}%
\contentsline {subsection}{\numberline {24.11.4}\textcolor {Chapter }{SumIntersectionMat}}{370}{subsection.24.11.4}%
\contentsline {subsection}{\numberline {24.11.5}\textcolor {Chapter }{BaseSteinitzVectors}}{370}{subsection.24.11.5}%
\contentsline {section}{\numberline {24.12}\textcolor {Chapter }{Triangular Matrices}}{371}{section.24.12}%
\contentsline {subsection}{\numberline {24.12.1}\textcolor {Chapter }{DiagonalOfMatrix}}{371}{subsection.24.12.1}%
\contentsline {subsection}{\numberline {24.12.2}\textcolor {Chapter }{UpperSubdiagonal}}{371}{subsection.24.12.2}%
\contentsline {subsection}{\numberline {24.12.3}\textcolor {Chapter }{DepthOfUpperTriangularMatrix}}{371}{subsection.24.12.3}%
\contentsline {section}{\numberline {24.13}\textcolor {Chapter }{Matrices as Linear Mappings}}{371}{section.24.13}%
\contentsline {subsection}{\numberline {24.13.1}\textcolor {Chapter }{CharacteristicPolynomial}}{371}{subsection.24.13.1}%
\contentsline {subsection}{\numberline {24.13.2}\textcolor {Chapter }{RationalCanonicalFormTransform}}{372}{subsection.24.13.2}%
\contentsline {subsection}{\numberline {24.13.3}\textcolor {Chapter }{JordanDecomposition}}{372}{subsection.24.13.3}%
\contentsline {subsection}{\numberline {24.13.4}\textcolor {Chapter }{BlownUpMat}}{373}{subsection.24.13.4}%
\contentsline {subsection}{\numberline {24.13.5}\textcolor {Chapter }{BlownUpVector}}{373}{subsection.24.13.5}%
\contentsline {subsection}{\numberline {24.13.6}\textcolor {Chapter }{CompanionMatrix}}{373}{subsection.24.13.6}%
\contentsline {section}{\numberline {24.14}\textcolor {Chapter }{Matrices over Finite Fields}}{374}{section.24.14}%
\contentsline {subsection}{\numberline {24.14.1}\textcolor {Chapter }{ImmutableMatrix}}{375}{subsection.24.14.1}%
\contentsline {subsection}{\numberline {24.14.2}\textcolor {Chapter }{ConvertToMatrixRep (for a list (and a field))}}{376}{subsection.24.14.2}%
\contentsline {subsection}{\numberline {24.14.3}\textcolor {Chapter }{ProjectiveOrder}}{376}{subsection.24.14.3}%
\contentsline {subsection}{\numberline {24.14.4}\textcolor {Chapter }{SimultaneousEigenvalues}}{376}{subsection.24.14.4}%
\contentsline {section}{\numberline {24.15}\textcolor {Chapter }{Inverse and Nullspace of an Integer Matrix Modulo an Ideal}}{377}{section.24.15}%
\contentsline {subsection}{\numberline {24.15.1}\textcolor {Chapter }{InverseMatMod}}{377}{subsection.24.15.1}%
\contentsline {subsection}{\numberline {24.15.2}\textcolor {Chapter }{BasisNullspaceModN}}{377}{subsection.24.15.2}%
\contentsline {subsection}{\numberline {24.15.3}\textcolor {Chapter }{NullspaceModQ}}{377}{subsection.24.15.3}%
\contentsline {section}{\numberline {24.16}\textcolor {Chapter }{Special Multiplication Algorithms for Matrices over GF(2)}}{378}{section.24.16}%
\contentsline {subsection}{\numberline {24.16.1}\textcolor {Chapter }{PROD{\textunderscore }GF2MAT{\textunderscore }GF2MAT{\textunderscore }SIMPLE}}{378}{subsection.24.16.1}%
\contentsline {subsection}{\numberline {24.16.2}\textcolor {Chapter }{PROD{\textunderscore }GF2MAT{\textunderscore }GF2MAT{\textunderscore }ADVANCED}}{378}{subsection.24.16.2}%
\contentsline {section}{\numberline {24.17}\textcolor {Chapter }{Block Matrices}}{378}{section.24.17}%
\contentsline {subsection}{\numberline {24.17.1}\textcolor {Chapter }{AsBlockMatrix}}{378}{subsection.24.17.1}%
\contentsline {subsection}{\numberline {24.17.2}\textcolor {Chapter }{BlockMatrix}}{379}{subsection.24.17.2}%
\contentsline {subsection}{\numberline {24.17.3}\textcolor {Chapter }{MatrixByBlockMatrix}}{379}{subsection.24.17.3}%
\contentsline {section}{\numberline {24.18}\textcolor {Chapter }{Linear Programming}}{379}{section.24.18}%
\contentsline {subsection}{\numberline {24.18.1}\textcolor {Chapter }{SimplexMethod}}{379}{subsection.24.18.1}%
\contentsline {chapter}{\numberline {25}\textcolor {Chapter }{Integral matrices and lattices}}{380}{chapter.25}%
\contentsline {section}{\numberline {25.1}\textcolor {Chapter }{Linear equations over the integers and Integral Matrices}}{380}{section.25.1}%
\contentsline {subsection}{\numberline {25.1.1}\textcolor {Chapter }{NullspaceIntMat}}{380}{subsection.25.1.1}%
\contentsline {subsection}{\numberline {25.1.2}\textcolor {Chapter }{SolutionIntMat}}{380}{subsection.25.1.2}%
\contentsline {subsection}{\numberline {25.1.3}\textcolor {Chapter }{SolutionNullspaceIntMat}}{380}{subsection.25.1.3}%
\contentsline {subsection}{\numberline {25.1.4}\textcolor {Chapter }{BaseIntMat}}{381}{subsection.25.1.4}%
\contentsline {subsection}{\numberline {25.1.5}\textcolor {Chapter }{BaseIntersectionIntMats}}{381}{subsection.25.1.5}%
\contentsline {subsection}{\numberline {25.1.6}\textcolor {Chapter }{ComplementIntMat}}{381}{subsection.25.1.6}%
\contentsline {section}{\numberline {25.2}\textcolor {Chapter }{Normal Forms over the Integers}}{382}{section.25.2}%
\contentsline {subsection}{\numberline {25.2.1}\textcolor {Chapter }{TriangulizedIntegerMat}}{382}{subsection.25.2.1}%
\contentsline {subsection}{\numberline {25.2.2}\textcolor {Chapter }{TriangulizedIntegerMatTransform}}{382}{subsection.25.2.2}%
\contentsline {subsection}{\numberline {25.2.3}\textcolor {Chapter }{TriangulizeIntegerMat}}{382}{subsection.25.2.3}%
\contentsline {subsection}{\numberline {25.2.4}\textcolor {Chapter }{HermiteNormalFormIntegerMat}}{383}{subsection.25.2.4}%
\contentsline {subsection}{\numberline {25.2.5}\textcolor {Chapter }{HermiteNormalFormIntegerMatTransform}}{383}{subsection.25.2.5}%
\contentsline {subsection}{\numberline {25.2.6}\textcolor {Chapter }{SmithNormalFormIntegerMat}}{383}{subsection.25.2.6}%
\contentsline {subsection}{\numberline {25.2.7}\textcolor {Chapter }{SmithNormalFormIntegerMatTransforms}}{383}{subsection.25.2.7}%
\contentsline {subsection}{\numberline {25.2.8}\textcolor {Chapter }{DiagonalizeIntMat}}{384}{subsection.25.2.8}%
\contentsline {subsection}{\numberline {25.2.9}\textcolor {Chapter }{NormalFormIntMat}}{384}{subsection.25.2.9}%
\contentsline {subsection}{\numberline {25.2.10}\textcolor {Chapter }{AbelianInvariantsOfList}}{385}{subsection.25.2.10}%
\contentsline {section}{\numberline {25.3}\textcolor {Chapter }{Determinant of an integer matrix}}{385}{section.25.3}%
\contentsline {subsection}{\numberline {25.3.1}\textcolor {Chapter }{DeterminantIntMat}}{385}{subsection.25.3.1}%
\contentsline {section}{\numberline {25.4}\textcolor {Chapter }{Decompositions}}{385}{section.25.4}%
\contentsline {subsection}{\numberline {25.4.1}\textcolor {Chapter }{Decomposition}}{386}{subsection.25.4.1}%
\contentsline {subsection}{\numberline {25.4.2}\textcolor {Chapter }{LinearIndependentColumns}}{386}{subsection.25.4.2}%
\contentsline {subsection}{\numberline {25.4.3}\textcolor {Chapter }{PadicCoefficients}}{387}{subsection.25.4.3}%
\contentsline {subsection}{\numberline {25.4.4}\textcolor {Chapter }{IntegralizedMat}}{387}{subsection.25.4.4}%
\contentsline {subsection}{\numberline {25.4.5}\textcolor {Chapter }{DecompositionInt}}{387}{subsection.25.4.5}%
\contentsline {section}{\numberline {25.5}\textcolor {Chapter }{Lattice Reduction}}{387}{section.25.5}%
\contentsline {subsection}{\numberline {25.5.1}\textcolor {Chapter }{LLLReducedBasis}}{387}{subsection.25.5.1}%
\contentsline {subsection}{\numberline {25.5.2}\textcolor {Chapter }{LLLReducedGramMat}}{388}{subsection.25.5.2}%
\contentsline {section}{\numberline {25.6}\textcolor {Chapter }{Orthogonal Embeddings}}{389}{section.25.6}%
\contentsline {subsection}{\numberline {25.6.1}\textcolor {Chapter }{OrthogonalEmbeddings}}{389}{subsection.25.6.1}%
\contentsline {subsection}{\numberline {25.6.2}\textcolor {Chapter }{ShortestVectors}}{389}{subsection.25.6.2}%
\contentsline {chapter}{\numberline {26}\textcolor {Chapter }{Vector and Matrix Objects}}{391}{chapter.26}%
\contentsline {section}{\numberline {26.1}\textcolor {Chapter }{Concepts and Rules for Vector and Matrix Objects}}{391}{section.26.1}%
\contentsline {section}{\numberline {26.2}\textcolor {Chapter }{Categories of Vector and Matrix Objects}}{392}{section.26.2}%
\contentsline {subsection}{\numberline {26.2.1}\textcolor {Chapter }{IsVectorObj}}{393}{subsection.26.2.1}%
\contentsline {subsection}{\numberline {26.2.2}\textcolor {Chapter }{IsMatrixObj}}{393}{subsection.26.2.2}%
\contentsline {subsection}{\numberline {26.2.3}\textcolor {Chapter }{IsMatrixOrMatrixObj}}{394}{subsection.26.2.3}%
\contentsline {subsection}{\numberline {26.2.4}\textcolor {Chapter }{IsRowListMatrix}}{394}{subsection.26.2.4}%
\contentsline {section}{\numberline {26.3}\textcolor {Chapter }{Defining Attributes of Vector and Matrix Objects}}{395}{section.26.3}%
\contentsline {subsection}{\numberline {26.3.1}\textcolor {Chapter }{BaseDomain}}{395}{subsection.26.3.1}%
\contentsline {subsection}{\numberline {26.3.2}\textcolor {Chapter }{ConstructingFilter}}{395}{subsection.26.3.2}%
\contentsline {subsection}{\numberline {26.3.3}\textcolor {Chapter }{CompatibleVectorFilter}}{395}{subsection.26.3.3}%
\contentsline {subsection}{\numberline {26.3.4}\textcolor {Chapter }{Length (for a vector object)}}{395}{subsection.26.3.4}%
\contentsline {subsection}{\numberline {26.3.5}\textcolor {Chapter }{NumberRows and NumberColumns}}{395}{subsection.26.3.5}%
\contentsline {section}{\numberline {26.4}\textcolor {Chapter }{Constructing Vector and Matrix Objects}}{396}{section.26.4}%
\contentsline {subsection}{\numberline {26.4.1}\textcolor {Chapter }{NewVector and NewZeroVector}}{396}{subsection.26.4.1}%
\contentsline {subsection}{\numberline {26.4.2}\textcolor {Chapter }{Vector}}{396}{subsection.26.4.2}%
\contentsline {subsection}{\numberline {26.4.3}\textcolor {Chapter }{ZeroVector}}{397}{subsection.26.4.3}%
\contentsline {subsection}{\numberline {26.4.4}\textcolor {Chapter }{NewMatrix, NewZeroMatrix, NewIdentityMatrix}}{397}{subsection.26.4.4}%
\contentsline {subsection}{\numberline {26.4.5}\textcolor {Chapter }{Matrix}}{397}{subsection.26.4.5}%
\contentsline {subsection}{\numberline {26.4.6}\textcolor {Chapter }{ZeroMatrix}}{398}{subsection.26.4.6}%
\contentsline {subsection}{\numberline {26.4.7}\textcolor {Chapter }{IdentityMatrix}}{398}{subsection.26.4.7}%
\contentsline {section}{\numberline {26.5}\textcolor {Chapter }{Operations for Base Domains of Vector and Matrix Objects}}{399}{section.26.5}%
\contentsline {subsection}{\numberline {26.5.1}\textcolor {Chapter }{OneOfBaseDomain and ZeroOfBaseDomain}}{399}{subsection.26.5.1}%
\contentsline {section}{\numberline {26.6}\textcolor {Chapter }{Operations for Vector and Matrix Objects}}{399}{section.26.6}%
\contentsline {subsection}{\numberline {26.6.1}\textcolor {Chapter }{Comparison of Vector and Matrix Objects}}{399}{subsection.26.6.1}%
\contentsline {subsection}{\numberline {26.6.2}\textcolor {Chapter }{Unpack}}{400}{subsection.26.6.2}%
\contentsline {subsection}{\numberline {26.6.3}\textcolor {Chapter }{ChangedBaseDomain}}{400}{subsection.26.6.3}%
\contentsline {subsection}{\numberline {26.6.4}\textcolor {Chapter }{Randomize}}{400}{subsection.26.6.4}%
\contentsline {section}{\numberline {26.7}\textcolor {Chapter }{List Like Operations for Vector Objects}}{400}{section.26.7}%
\contentsline {subsection}{\numberline {26.7.1}\textcolor {Chapter }{Element Access and Assignment for Vector Objects}}{400}{subsection.26.7.1}%
\contentsline {subsection}{\numberline {26.7.2}\textcolor {Chapter }{PositionNonZero (for a vector object)}}{401}{subsection.26.7.2}%
\contentsline {subsection}{\numberline {26.7.3}\textcolor {Chapter }{PositionLastNonZero (for a vector object)}}{401}{subsection.26.7.3}%
\contentsline {subsection}{\numberline {26.7.4}\textcolor {Chapter }{ListOp (for vector object and function)}}{401}{subsection.26.7.4}%
\contentsline {section}{\numberline {26.8}\textcolor {Chapter }{Arithmetical Operations for Vector Objects}}{401}{section.26.8}%
\contentsline {subsection}{\numberline {26.8.1}\textcolor {Chapter }{Unary Arithmetical Operations for Vector Objects}}{401}{subsection.26.8.1}%
\contentsline {subsection}{\numberline {26.8.2}\textcolor {Chapter }{Binary Arithmetical Operations for Vector Objects}}{402}{subsection.26.8.2}%
\contentsline {subsection}{\numberline {26.8.3}\textcolor {Chapter }{AddVector (for two vector objects)}}{402}{subsection.26.8.3}%
\contentsline {subsection}{\numberline {26.8.4}\textcolor {Chapter }{MultVector (for a vector object)}}{403}{subsection.26.8.4}%
\contentsline {section}{\numberline {26.9}\textcolor {Chapter }{Operations for Vector Objects}}{403}{section.26.9}%
\contentsline {subsection}{\numberline {26.9.1}\textcolor {Chapter }{ConcatenationOfVectors}}{403}{subsection.26.9.1}%
\contentsline {subsection}{\numberline {26.9.2}\textcolor {Chapter }{ExtractSubVector}}{403}{subsection.26.9.2}%
\contentsline {subsection}{\numberline {26.9.3}\textcolor {Chapter }{CopySubVector}}{403}{subsection.26.9.3}%
\contentsline {subsection}{\numberline {26.9.4}\textcolor {Chapter }{WeightOfVector (for a vector object)}}{404}{subsection.26.9.4}%
\contentsline {subsection}{\numberline {26.9.5}\textcolor {Chapter }{DistanceOfVectors (for two vector objects)}}{404}{subsection.26.9.5}%
\contentsline {section}{\numberline {26.10}\textcolor {Chapter }{Arithmetical Operations for Matrix Objects}}{404}{section.26.10}%
\contentsline {subsection}{\numberline {26.10.1}\textcolor {Chapter }{Unary Arithmetical Operations for Matrix Objects}}{404}{subsection.26.10.1}%
\contentsline {subsection}{\numberline {26.10.2}\textcolor {Chapter }{Binary Arithmetical Operations for Matrix Objects}}{405}{subsection.26.10.2}%
\contentsline {section}{\numberline {26.11}\textcolor {Chapter }{Operations for Matrix Objects}}{405}{section.26.11}%
\contentsline {subsection}{\numberline {26.11.1}\textcolor {Chapter }{MatElm}}{405}{subsection.26.11.1}%
\contentsline {subsection}{\numberline {26.11.2}\textcolor {Chapter }{SetMatElm}}{405}{subsection.26.11.2}%
\contentsline {subsection}{\numberline {26.11.3}\textcolor {Chapter }{ExtractSubMatrix}}{406}{subsection.26.11.3}%
\contentsline {subsection}{\numberline {26.11.4}\textcolor {Chapter }{MutableCopyMatrix (for a matrix object)}}{406}{subsection.26.11.4}%
\contentsline {subsection}{\numberline {26.11.5}\textcolor {Chapter }{CopySubMatrix}}{406}{subsection.26.11.5}%
\contentsline {subsection}{\numberline {26.11.6}\textcolor {Chapter }{CompatibleVector (for a matrix object)}}{406}{subsection.26.11.6}%
\contentsline {subsection}{\numberline {26.11.7}\textcolor {Chapter }{RowsOfMatrix (for a matrix object)}}{406}{subsection.26.11.7}%
\contentsline {subsection}{\numberline {26.11.8}\textcolor {Chapter }{CompanionMatrix}}{407}{subsection.26.11.8}%
\contentsline {section}{\numberline {26.12}\textcolor {Chapter }{Operations for Row List Matrix Objects}}{407}{section.26.12}%
\contentsline {subsection}{\numberline {26.12.1}\textcolor {Chapter }{List Access for a Row List Matrix}}{407}{subsection.26.12.1}%
\contentsline {subsection}{\numberline {26.12.2}\textcolor {Chapter }{List Assignment for a Row List Matrix}}{408}{subsection.26.12.2}%
\contentsline {subsection}{\numberline {26.12.3}\textcolor {Chapter }{Sublist Access for a Row List Matrix}}{408}{subsection.26.12.3}%
\contentsline {subsection}{\numberline {26.12.4}\textcolor {Chapter }{Sublist Assignment for a Row List Matrix}}{408}{subsection.26.12.4}%
\contentsline {subsection}{\numberline {26.12.5}\textcolor {Chapter }{IsBound\texttt {\symbol {92}}[\texttt {\symbol {92}}] (for a row list matrix)}}{408}{subsection.26.12.5}%
\contentsline {subsection}{\numberline {26.12.6}\textcolor {Chapter }{Unbind\texttt {\symbol {92}}[\texttt {\symbol {92}}] (for a row list matrix)}}{408}{subsection.26.12.6}%
\contentsline {subsection}{\numberline {26.12.7}\textcolor {Chapter }{Add (for a row list matrix and a vector object)}}{408}{subsection.26.12.7}%
\contentsline {subsection}{\numberline {26.12.8}\textcolor {Chapter }{Remove (for a row list matrix)}}{409}{subsection.26.12.8}%
\contentsline {subsection}{\numberline {26.12.9}\textcolor {Chapter }{Append (for two row list matrices)}}{409}{subsection.26.12.9}%
\contentsline {subsection}{\numberline {26.12.10}\textcolor {Chapter }{ShallowCopy (for a row list matrix)}}{409}{subsection.26.12.10}%
\contentsline {subsection}{\numberline {26.12.11}\textcolor {Chapter }{ListOp (for a row list matrix)}}{409}{subsection.26.12.11}%
\contentsline {section}{\numberline {26.13}\textcolor {Chapter }{Basic operations for row/column reductions}}{409}{section.26.13}%
\contentsline {subsection}{\numberline {26.13.1}\textcolor {Chapter }{MultMatrixRowLeft}}{409}{subsection.26.13.1}%
\contentsline {subsection}{\numberline {26.13.2}\textcolor {Chapter }{MultMatrixRowRight}}{410}{subsection.26.13.2}%
\contentsline {subsection}{\numberline {26.13.3}\textcolor {Chapter }{MultMatrixColumnRight}}{410}{subsection.26.13.3}%
\contentsline {subsection}{\numberline {26.13.4}\textcolor {Chapter }{MultMatrixColumnLeft}}{410}{subsection.26.13.4}%
\contentsline {subsection}{\numberline {26.13.5}\textcolor {Chapter }{AddMatrixRowsLeft}}{410}{subsection.26.13.5}%
\contentsline {subsection}{\numberline {26.13.6}\textcolor {Chapter }{AddMatrixRowsRight}}{410}{subsection.26.13.6}%
\contentsline {subsection}{\numberline {26.13.7}\textcolor {Chapter }{AddMatrixColumnsRight}}{410}{subsection.26.13.7}%
\contentsline {subsection}{\numberline {26.13.8}\textcolor {Chapter }{AddMatrixColumnsLeft}}{411}{subsection.26.13.8}%
\contentsline {subsection}{\numberline {26.13.9}\textcolor {Chapter }{PositionNonZeroInRow}}{411}{subsection.26.13.9}%
\contentsline {subsection}{\numberline {26.13.10}\textcolor {Chapter }{SwapMatrixRows}}{411}{subsection.26.13.10}%
\contentsline {subsection}{\numberline {26.13.11}\textcolor {Chapter }{SwapMatrixColumns}}{411}{subsection.26.13.11}%
\contentsline {subsection}{\numberline {26.13.12}\textcolor {Chapter }{AddMatrix}}{411}{subsection.26.13.12}%
\contentsline {subsection}{\numberline {26.13.13}\textcolor {Chapter }{MultMatrix}}{412}{subsection.26.13.13}%
\contentsline {section}{\numberline {26.14}\textcolor {Chapter }{Implementing New Vector and Matrix Objects Types}}{412}{section.26.14}%
\contentsline {section}{\numberline {26.15}\textcolor {Chapter }{Available Representations of Vector Objects}}{413}{section.26.15}%
\contentsline {subsection}{\numberline {26.15.1}\textcolor {Chapter }{IsGF2VectorRep}}{414}{subsection.26.15.1}%
\contentsline {subsection}{\numberline {26.15.2}\textcolor {Chapter }{Is8BitVectorRep}}{414}{subsection.26.15.2}%
\contentsline {subsection}{\numberline {26.15.3}\textcolor {Chapter }{IsPlistVectorRep}}{414}{subsection.26.15.3}%
\contentsline {subsection}{\numberline {26.15.4}\textcolor {Chapter }{IsZmodnZVectorRep}}{414}{subsection.26.15.4}%
\contentsline {section}{\numberline {26.16}\textcolor {Chapter }{Available Representations of Matrix Objects}}{414}{section.26.16}%
\contentsline {subsection}{\numberline {26.16.1}\textcolor {Chapter }{IsGF2MatrixRep}}{414}{subsection.26.16.1}%
\contentsline {subsection}{\numberline {26.16.2}\textcolor {Chapter }{Is8BitMatrixRep}}{415}{subsection.26.16.2}%
\contentsline {subsection}{\numberline {26.16.3}\textcolor {Chapter }{IsPlistMatrixRep}}{415}{subsection.26.16.3}%
\contentsline {subsection}{\numberline {26.16.4}\textcolor {Chapter }{IsGenericMatrixRep}}{415}{subsection.26.16.4}%
\contentsline {subsection}{\numberline {26.16.5}\textcolor {Chapter }{IsZmodnZMatrixRep}}{416}{subsection.26.16.5}%
\contentsline {section}{\numberline {26.17}\textcolor {Chapter }{Groups Consisting of Matrix Objects}}{416}{section.26.17}%
\contentsline {subsection}{\numberline {26.17.1}\textcolor {Chapter }{ConstructingFiltersForMatrixGroupElements}}{416}{subsection.26.17.1}%
\contentsline {section}{\numberline {26.18}\textcolor {Chapter }{How to Write Code for Vector and Matrix Objects}}{416}{section.26.18}%
\contentsline {chapter}{\numberline {27}\textcolor {Chapter }{Strings and Characters}}{419}{chapter.27}%
\contentsline {section}{\numberline {27.1}\textcolor {Chapter }{IsChar and IsString}}{419}{section.27.1}%
\contentsline {subsection}{\numberline {27.1.1}\textcolor {Chapter }{IsChar}}{419}{subsection.27.1.1}%
\contentsline {subsection}{\numberline {27.1.2}\textcolor {Chapter }{IsString}}{419}{subsection.27.1.2}%
\contentsline {subsection}{\numberline {27.1.3}\textcolor {Chapter }{Strings As Lists}}{420}{subsection.27.1.3}%
\contentsline {subsection}{\numberline {27.1.4}\textcolor {Chapter }{Printing Strings}}{420}{subsection.27.1.4}%
\contentsline {section}{\numberline {27.2}\textcolor {Chapter }{Special Characters}}{421}{section.27.2}%
\contentsline {section}{\numberline {27.3}\textcolor {Chapter }{Triple Quoted Strings}}{423}{section.27.3}%
\contentsline {section}{\numberline {27.4}\textcolor {Chapter }{Internally Represented Strings}}{423}{section.27.4}%
\contentsline {subsection}{\numberline {27.4.1}\textcolor {Chapter }{IsStringRep}}{423}{subsection.27.4.1}%
\contentsline {subsection}{\numberline {27.4.2}\textcolor {Chapter }{ConvertToStringRep}}{423}{subsection.27.4.2}%
\contentsline {subsection}{\numberline {27.4.3}\textcolor {Chapter }{CopyToStringRep}}{423}{subsection.27.4.3}%
\contentsline {subsection}{\numberline {27.4.4}\textcolor {Chapter }{IsEmptyString}}{424}{subsection.27.4.4}%
\contentsline {subsection}{\numberline {27.4.5}\textcolor {Chapter }{EmptyString}}{424}{subsection.27.4.5}%
\contentsline {subsection}{\numberline {27.4.6}\textcolor {Chapter }{CharsFamily}}{424}{subsection.27.4.6}%
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\contentsline {chapter}{\numberline {33}\textcolor {Chapter }{Relations}}{516}{chapter.33}%
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\contentsline {chapter}{\numberline {36}\textcolor {Chapter }{Words}}{542}{chapter.36}%
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\contentsline {subsection}{\numberline {39.15.23}\textcolor {Chapter }{PrimePGroup}}{610}{subsection.39.15.23}%
\contentsline {subsection}{\numberline {39.15.24}\textcolor {Chapter }{PClassPGroup}}{611}{subsection.39.15.24}%
\contentsline {subsection}{\numberline {39.15.25}\textcolor {Chapter }{RankPGroup}}{611}{subsection.39.15.25}%
\contentsline {subsection}{\numberline {39.15.26}\textcolor {Chapter }{IsPSolvable}}{611}{subsection.39.15.26}%
\contentsline {subsection}{\numberline {39.15.27}\textcolor {Chapter }{IsPNilpotent}}{611}{subsection.39.15.27}%
\contentsline {section}{\numberline {39.16}\textcolor {Chapter }{Numerical Group Attributes}}{611}{section.39.16}%
\contentsline {subsection}{\numberline {39.16.1}\textcolor {Chapter }{AbelianInvariants}}{612}{subsection.39.16.1}%
\contentsline {subsection}{\numberline {39.16.2}\textcolor {Chapter }{Exponent}}{612}{subsection.39.16.2}%
\contentsline {subsection}{\numberline {39.16.3}\textcolor {Chapter }{EulerianFunction}}{612}{subsection.39.16.3}%
\contentsline {section}{\numberline {39.17}\textcolor {Chapter }{Subgroup Series}}{612}{section.39.17}%
\contentsline {subsection}{\numberline {39.17.1}\textcolor {Chapter }{ChiefSeries}}{613}{subsection.39.17.1}%
\contentsline {subsection}{\numberline {39.17.2}\textcolor {Chapter }{ChiefSeriesThrough}}{613}{subsection.39.17.2}%
\contentsline {subsection}{\numberline {39.17.3}\textcolor {Chapter }{ChiefSeriesUnderAction}}{613}{subsection.39.17.3}%
\contentsline {subsection}{\numberline {39.17.4}\textcolor {Chapter }{SubnormalSeries}}{613}{subsection.39.17.4}%
\contentsline {subsection}{\numberline {39.17.5}\textcolor {Chapter }{CompositionSeries}}{614}{subsection.39.17.5}%
\contentsline {subsection}{\numberline {39.17.6}\textcolor {Chapter }{DisplayCompositionSeries}}{614}{subsection.39.17.6}%
\contentsline {subsection}{\numberline {39.17.7}\textcolor {Chapter }{DerivedSeriesOfGroup}}{614}{subsection.39.17.7}%
\contentsline {subsection}{\numberline {39.17.8}\textcolor {Chapter }{DerivedLength}}{614}{subsection.39.17.8}%
\contentsline {subsection}{\numberline {39.17.9}\textcolor {Chapter }{ElementaryAbelianSeries}}{615}{subsection.39.17.9}%
\contentsline {subsection}{\numberline {39.17.10}\textcolor {Chapter }{InvariantElementaryAbelianSeries}}{615}{subsection.39.17.10}%
\contentsline {subsection}{\numberline {39.17.11}\textcolor {Chapter }{LowerCentralSeriesOfGroup}}{615}{subsection.39.17.11}%
\contentsline {subsection}{\numberline {39.17.12}\textcolor {Chapter }{UpperCentralSeriesOfGroup}}{615}{subsection.39.17.12}%
\contentsline {subsection}{\numberline {39.17.13}\textcolor {Chapter }{PCentralSeries}}{616}{subsection.39.17.13}%
\contentsline {subsection}{\numberline {39.17.14}\textcolor {Chapter }{JenningsSeries}}{616}{subsection.39.17.14}%
\contentsline {subsection}{\numberline {39.17.15}\textcolor {Chapter }{DimensionsLoewyFactors}}{616}{subsection.39.17.15}%
\contentsline {subsection}{\numberline {39.17.16}\textcolor {Chapter }{AscendingChain}}{616}{subsection.39.17.16}%
\contentsline {subsection}{\numberline {39.17.17}\textcolor {Chapter }{IntermediateGroup}}{617}{subsection.39.17.17}%
\contentsline {subsection}{\numberline {39.17.18}\textcolor {Chapter }{IntermediateSubgroups}}{617}{subsection.39.17.18}%
\contentsline {subsection}{\numberline {39.17.19}\textcolor {Chapter }{StructuralSeriesOfGroup}}{617}{subsection.39.17.19}%
\contentsline {section}{\numberline {39.18}\textcolor {Chapter }{Factor Groups}}{617}{section.39.18}%
\contentsline {subsection}{\numberline {39.18.1}\textcolor {Chapter }{NaturalHomomorphismByNormalSubgroup}}{617}{subsection.39.18.1}%
\contentsline {subsection}{\numberline {39.18.2}\textcolor {Chapter }{FactorGroup}}{618}{subsection.39.18.2}%
\contentsline {subsection}{\numberline {39.18.3}\textcolor {Chapter }{CommutatorFactorGroup}}{618}{subsection.39.18.3}%
\contentsline {subsection}{\numberline {39.18.4}\textcolor {Chapter }{MaximalAbelianQuotient}}{618}{subsection.39.18.4}%
\contentsline {subsection}{\numberline {39.18.5}\textcolor {Chapter }{HasAbelianFactorGroup}}{618}{subsection.39.18.5}%
\contentsline {subsection}{\numberline {39.18.6}\textcolor {Chapter }{HasElementaryAbelianFactorGroup}}{619}{subsection.39.18.6}%
\contentsline {subsection}{\numberline {39.18.7}\textcolor {Chapter }{CentralizerModulo}}{619}{subsection.39.18.7}%
\contentsline {section}{\numberline {39.19}\textcolor {Chapter }{Sets of Subgroups}}{619}{section.39.19}%
\contentsline {subsection}{\numberline {39.19.1}\textcolor {Chapter }{ConjugacyClassSubgroups}}{619}{subsection.39.19.1}%
\contentsline {subsection}{\numberline {39.19.2}\textcolor {Chapter }{IsConjugacyClassSubgroupsRep}}{619}{subsection.39.19.2}%
\contentsline {subsection}{\numberline {39.19.3}\textcolor {Chapter }{ConjugacyClassesSubgroups}}{620}{subsection.39.19.3}%
\contentsline {subsection}{\numberline {39.19.4}\textcolor {Chapter }{ConjugacyClassesMaximalSubgroups}}{620}{subsection.39.19.4}%
\contentsline {subsection}{\numberline {39.19.5}\textcolor {Chapter }{MaximalSubgroupClassReps}}{620}{subsection.39.19.5}%
\contentsline {subsection}{\numberline {39.19.6}\textcolor {Chapter }{LowIndexSubgroups}}{620}{subsection.39.19.6}%
\contentsline {subsection}{\numberline {39.19.7}\textcolor {Chapter }{AllSubgroups}}{621}{subsection.39.19.7}%
\contentsline {subsection}{\numberline {39.19.8}\textcolor {Chapter }{MaximalSubgroups}}{621}{subsection.39.19.8}%
\contentsline {subsection}{\numberline {39.19.9}\textcolor {Chapter }{NormalSubgroups}}{621}{subsection.39.19.9}%
\contentsline {subsection}{\numberline {39.19.10}\textcolor {Chapter }{MaximalNormalSubgroups}}{622}{subsection.39.19.10}%
\contentsline {subsection}{\numberline {39.19.11}\textcolor {Chapter }{MinimalNormalSubgroups}}{622}{subsection.39.19.11}%
\contentsline {subsection}{\numberline {39.19.12}\textcolor {Chapter }{CharacteristicSubgroups}}{622}{subsection.39.19.12}%
\contentsline {section}{\numberline {39.20}\textcolor {Chapter }{Subgroup Lattice}}{622}{section.39.20}%
\contentsline {subsection}{\numberline {39.20.1}\textcolor {Chapter }{LatticeSubgroups}}{622}{subsection.39.20.1}%
\contentsline {subsection}{\numberline {39.20.2}\textcolor {Chapter }{ClassElementLattice}}{623}{subsection.39.20.2}%
\contentsline {subsection}{\numberline {39.20.3}\textcolor {Chapter }{DotFileLatticeSubgroups}}{623}{subsection.39.20.3}%
\contentsline {subsection}{\numberline {39.20.4}\textcolor {Chapter }{MaximalSubgroupsLattice}}{623}{subsection.39.20.4}%
\contentsline {subsection}{\numberline {39.20.5}\textcolor {Chapter }{MinimalSupergroupsLattice}}{624}{subsection.39.20.5}%
\contentsline {subsection}{\numberline {39.20.6}\textcolor {Chapter }{LowLayerSubgroups}}{625}{subsection.39.20.6}%
\contentsline {subsection}{\numberline {39.20.7}\textcolor {Chapter }{ContainedConjugates}}{625}{subsection.39.20.7}%
\contentsline {subsection}{\numberline {39.20.8}\textcolor {Chapter }{ContainingConjugates}}{625}{subsection.39.20.8}%
\contentsline {subsection}{\numberline {39.20.9}\textcolor {Chapter }{MinimalFaithfulPermutationDegree}}{626}{subsection.39.20.9}%
\contentsline {subsection}{\numberline {39.20.10}\textcolor {Chapter }{RepresentativesPerfectSubgroups}}{626}{subsection.39.20.10}%
\contentsline {subsection}{\numberline {39.20.11}\textcolor {Chapter }{ConjugacyClassesPerfectSubgroups}}{626}{subsection.39.20.11}%
\contentsline {subsection}{\numberline {39.20.12}\textcolor {Chapter }{Zuppos}}{627}{subsection.39.20.12}%
\contentsline {subsection}{\numberline {39.20.13}\textcolor {Chapter }{InfoLattice}}{627}{subsection.39.20.13}%
\contentsline {section}{\numberline {39.21}\textcolor {Chapter }{Specific Methods for Subgroup Lattice Computations}}{627}{section.39.21}%
\contentsline {subsection}{\numberline {39.21.1}\textcolor {Chapter }{LatticeByCyclicExtension}}{627}{subsection.39.21.1}%
\contentsline {subsection}{\numberline {39.21.2}\textcolor {Chapter }{InvariantSubgroupsElementaryAbelianGroup}}{628}{subsection.39.21.2}%
\contentsline {subsection}{\numberline {39.21.3}\textcolor {Chapter }{SubgroupsSolvableGroup}}{628}{subsection.39.21.3}%
\contentsline {subsection}{\numberline {39.21.4}\textcolor {Chapter }{SizeConsiderFunction}}{629}{subsection.39.21.4}%
\contentsline {subsection}{\numberline {39.21.5}\textcolor {Chapter }{ExactSizeConsiderFunction}}{630}{subsection.39.21.5}%
\contentsline {subsection}{\numberline {39.21.6}\textcolor {Chapter }{InfoPcSubgroup}}{630}{subsection.39.21.6}%
\contentsline {section}{\numberline {39.22}\textcolor {Chapter }{Special Generating Sets}}{630}{section.39.22}%
\contentsline {subsection}{\numberline {39.22.1}\textcolor {Chapter }{GeneratorsSmallest}}{630}{subsection.39.22.1}%
\contentsline {subsection}{\numberline {39.22.2}\textcolor {Chapter }{LargestElementGroup}}{631}{subsection.39.22.2}%
\contentsline {subsection}{\numberline {39.22.3}\textcolor {Chapter }{MinimalGeneratingSet}}{631}{subsection.39.22.3}%
\contentsline {subsection}{\numberline {39.22.4}\textcolor {Chapter }{SmallGeneratingSet}}{631}{subsection.39.22.4}%
\contentsline {subsection}{\numberline {39.22.5}\textcolor {Chapter }{IndependentGeneratorsOfAbelianGroup}}{631}{subsection.39.22.5}%
\contentsline {subsection}{\numberline {39.22.6}\textcolor {Chapter }{IndependentGeneratorExponents}}{632}{subsection.39.22.6}%
\contentsline {section}{\numberline {39.23}\textcolor {Chapter }{1\texttt {\symbol {45}}Cohomology}}{632}{section.39.23}%
\contentsline {subsection}{\numberline {39.23.1}\textcolor {Chapter }{OneCocycles}}{632}{subsection.39.23.1}%
\contentsline {subsection}{\numberline {39.23.2}\textcolor {Chapter }{OneCoboundaries}}{633}{subsection.39.23.2}%
\contentsline {subsection}{\numberline {39.23.3}\textcolor {Chapter }{OCOneCocycles}}{634}{subsection.39.23.3}%
\contentsline {subsection}{\numberline {39.23.4}\textcolor {Chapter }{ComplementClassesRepresentativesEA}}{634}{subsection.39.23.4}%
\contentsline {subsection}{\numberline {39.23.5}\textcolor {Chapter }{InfoCoh}}{635}{subsection.39.23.5}%
\contentsline {section}{\numberline {39.24}\textcolor {Chapter }{Schur Covers and Multipliers}}{635}{section.39.24}%
\contentsline {subsection}{\numberline {39.24.1}\textcolor {Chapter }{EpimorphismSchurCover}}{635}{subsection.39.24.1}%
\contentsline {subsection}{\numberline {39.24.2}\textcolor {Chapter }{SchurCover}}{635}{subsection.39.24.2}%
\contentsline {subsection}{\numberline {39.24.3}\textcolor {Chapter }{AbelianInvariantsMultiplier}}{636}{subsection.39.24.3}%
\contentsline {subsection}{\numberline {39.24.4}\textcolor {Chapter }{Epicentre}}{636}{subsection.39.24.4}%
\contentsline {subsection}{\numberline {39.24.5}\textcolor {Chapter }{NonabelianExteriorSquare}}{636}{subsection.39.24.5}%
\contentsline {subsection}{\numberline {39.24.6}\textcolor {Chapter }{EpimorphismNonabelianExteriorSquare}}{636}{subsection.39.24.6}%
\contentsline {subsection}{\numberline {39.24.7}\textcolor {Chapter }{IsCentralFactor}}{637}{subsection.39.24.7}%
\contentsline {subsection}{\numberline {39.24.8}\textcolor {Chapter }{Covering groups of symmetric groups}}{637}{subsection.39.24.8}%
\contentsline {subsection}{\numberline {39.24.9}\textcolor {Chapter }{BasicSpinRepresentationOfSymmetricGroup}}{637}{subsection.39.24.9}%
\contentsline {subsection}{\numberline {39.24.10}\textcolor {Chapter }{SchurCoverOfSymmetricGroup}}{638}{subsection.39.24.10}%
\contentsline {subsection}{\numberline {39.24.11}\textcolor {Chapter }{DoubleCoverOfAlternatingGroup}}{638}{subsection.39.24.11}%
\contentsline {section}{\numberline {39.25}\textcolor {Chapter }{2\texttt {\symbol {45}}Cohomology}}{638}{section.39.25}%
\contentsline {subsection}{\numberline {39.25.1}\textcolor {Chapter }{TwoCohomologyGeneric}}{638}{subsection.39.25.1}%
\contentsline {subsection}{\numberline {39.25.2}\textcolor {Chapter }{FpGroupCocycle}}{639}{subsection.39.25.2}%
\contentsline {section}{\numberline {39.26}\textcolor {Chapter }{Tests for the Availability of Methods}}{640}{section.39.26}%
\contentsline {subsection}{\numberline {39.26.1}\textcolor {Chapter }{CanEasilyTestMembership}}{640}{subsection.39.26.1}%
\contentsline {subsection}{\numberline {39.26.2}\textcolor {Chapter }{CanEasilyComputeWithIndependentGensAbelianGroup}}{640}{subsection.39.26.2}%
\contentsline {subsection}{\numberline {39.26.3}\textcolor {Chapter }{CanComputeSize}}{641}{subsection.39.26.3}%
\contentsline {subsection}{\numberline {39.26.4}\textcolor {Chapter }{CanComputeSizeAnySubgroup}}{641}{subsection.39.26.4}%
\contentsline {subsection}{\numberline {39.26.5}\textcolor {Chapter }{CanComputeIndex}}{641}{subsection.39.26.5}%
\contentsline {subsection}{\numberline {39.26.6}\textcolor {Chapter }{CanComputeIsSubset}}{641}{subsection.39.26.6}%
\contentsline {subsection}{\numberline {39.26.7}\textcolor {Chapter }{KnowsHowToDecompose}}{641}{subsection.39.26.7}%
\contentsline {section}{\numberline {39.27}\textcolor {Chapter }{Specific functions for Normalizer calculation}}{641}{section.39.27}%
\contentsline {subsection}{\numberline {39.27.1}\textcolor {Chapter }{NormalizerViaRadical}}{641}{subsection.39.27.1}%
\contentsline {chapter}{\numberline {40}\textcolor {Chapter }{Group Homomorphisms}}{643}{chapter.40}%
\contentsline {section}{\numberline {40.1}\textcolor {Chapter }{Creating Group Homomorphisms}}{643}{section.40.1}%
\contentsline {subsection}{\numberline {40.1.1}\textcolor {Chapter }{GroupHomomorphismByImages}}{643}{subsection.40.1.1}%
\contentsline {subsection}{\numberline {40.1.2}\textcolor {Chapter }{GroupHomomorphismByImagesNC}}{644}{subsection.40.1.2}%
\contentsline {subsection}{\numberline {40.1.3}\textcolor {Chapter }{GroupGeneralMappingByImages}}{644}{subsection.40.1.3}%
\contentsline {subsection}{\numberline {40.1.4}\textcolor {Chapter }{GroupHomomorphismByFunction}}{645}{subsection.40.1.4}%
\contentsline {subsection}{\numberline {40.1.5}\textcolor {Chapter }{AsGroupGeneralMappingByImages}}{645}{subsection.40.1.5}%
\contentsline {section}{\numberline {40.2}\textcolor {Chapter }{Operations for Group Homomorphisms}}{646}{section.40.2}%
\contentsline {section}{\numberline {40.3}\textcolor {Chapter }{Efficiency of Homomorphisms}}{647}{section.40.3}%
\contentsline {subsection}{\numberline {40.3.1}\textcolor {Chapter }{Mappings given on generators}}{647}{subsection.40.3.1}%
\contentsline {subsection}{\numberline {40.3.2}\textcolor {Chapter }{Action homomorphisms}}{647}{subsection.40.3.2}%
\contentsline {subsection}{\numberline {40.3.3}\textcolor {Chapter }{Mappings given by functions}}{648}{subsection.40.3.3}%
\contentsline {subsection}{\numberline {40.3.4}\textcolor {Chapter }{Other operations}}{648}{subsection.40.3.4}%
\contentsline {subsection}{\numberline {40.3.5}\textcolor {Chapter }{ImagesSmallestGenerators}}{648}{subsection.40.3.5}%
\contentsline {section}{\numberline {40.4}\textcolor {Chapter }{Homomorphism for very large groups}}{648}{section.40.4}%
\contentsline {section}{\numberline {40.5}\textcolor {Chapter }{Nice Monomorphisms}}{649}{section.40.5}%
\contentsline {subsection}{\numberline {40.5.1}\textcolor {Chapter }{IsHandledByNiceMonomorphism}}{649}{subsection.40.5.1}%
\contentsline {subsection}{\numberline {40.5.2}\textcolor {Chapter }{NiceMonomorphism}}{649}{subsection.40.5.2}%
\contentsline {subsection}{\numberline {40.5.3}\textcolor {Chapter }{NiceObject}}{650}{subsection.40.5.3}%
\contentsline {subsection}{\numberline {40.5.4}\textcolor {Chapter }{IsCanonicalNiceMonomorphism}}{650}{subsection.40.5.4}%
\contentsline {section}{\numberline {40.6}\textcolor {Chapter }{Group Automorphisms}}{650}{section.40.6}%
\contentsline {subsection}{\numberline {40.6.1}\textcolor {Chapter }{ConjugatorIsomorphism}}{650}{subsection.40.6.1}%
\contentsline {subsection}{\numberline {40.6.2}\textcolor {Chapter }{ConjugatorAutomorphism}}{650}{subsection.40.6.2}%
\contentsline {subsection}{\numberline {40.6.3}\textcolor {Chapter }{InnerAutomorphism}}{651}{subsection.40.6.3}%
\contentsline {subsection}{\numberline {40.6.4}\textcolor {Chapter }{IsConjugatorIsomorphism}}{651}{subsection.40.6.4}%
\contentsline {subsection}{\numberline {40.6.5}\textcolor {Chapter }{ConjugatorOfConjugatorIsomorphism}}{651}{subsection.40.6.5}%
\contentsline {section}{\numberline {40.7}\textcolor {Chapter }{Groups of Automorphisms}}{652}{section.40.7}%
\contentsline {subsection}{\numberline {40.7.1}\textcolor {Chapter }{AutomorphismGroup}}{652}{subsection.40.7.1}%
\contentsline {subsection}{\numberline {40.7.2}\textcolor {Chapter }{IsGroupOfAutomorphisms}}{652}{subsection.40.7.2}%
\contentsline {subsection}{\numberline {40.7.3}\textcolor {Chapter }{AutomorphismDomain}}{653}{subsection.40.7.3}%
\contentsline {subsection}{\numberline {40.7.4}\textcolor {Chapter }{IsAutomorphismGroup}}{653}{subsection.40.7.4}%
\contentsline {subsection}{\numberline {40.7.5}\textcolor {Chapter }{InnerAutomorphismsAutomorphismGroup}}{653}{subsection.40.7.5}%
\contentsline {subsection}{\numberline {40.7.6}\textcolor {Chapter }{InnerAutomorphismGroup}}{653}{subsection.40.7.6}%
\contentsline {subsection}{\numberline {40.7.7}\textcolor {Chapter }{InducedAutomorphism}}{654}{subsection.40.7.7}%
\contentsline {section}{\numberline {40.8}\textcolor {Chapter }{Calculating with Group Automorphisms}}{654}{section.40.8}%
\contentsline {subsection}{\numberline {40.8.1}\textcolor {Chapter }{AssignNiceMonomorphismAutomorphismGroup}}{654}{subsection.40.8.1}%
\contentsline {subsection}{\numberline {40.8.2}\textcolor {Chapter }{NiceMonomorphismAutomGroup}}{655}{subsection.40.8.2}%
\contentsline {section}{\numberline {40.9}\textcolor {Chapter }{Searching for Homomorphisms}}{655}{section.40.9}%
\contentsline {subsection}{\numberline {40.9.1}\textcolor {Chapter }{IsomorphismGroups}}{655}{subsection.40.9.1}%
\contentsline {subsection}{\numberline {40.9.2}\textcolor {Chapter }{AllHomomorphismClasses}}{655}{subsection.40.9.2}%
\contentsline {subsection}{\numberline {40.9.3}\textcolor {Chapter }{AllHomomorphisms}}{655}{subsection.40.9.3}%
\contentsline {subsection}{\numberline {40.9.4}\textcolor {Chapter }{GQuotients}}{656}{subsection.40.9.4}%
\contentsline {subsection}{\numberline {40.9.5}\textcolor {Chapter }{IsomorphicSubgroups}}{656}{subsection.40.9.5}%
\contentsline {subsection}{\numberline {40.9.6}\textcolor {Chapter }{MorClassLoop}}{657}{subsection.40.9.6}%
\contentsline {section}{\numberline {40.10}\textcolor {Chapter }{Representations for Group Homomorphisms}}{658}{section.40.10}%
\contentsline {subsection}{\numberline {40.10.1}\textcolor {Chapter }{IsGroupGeneralMappingByImages}}{658}{subsection.40.10.1}%
\contentsline {subsection}{\numberline {40.10.2}\textcolor {Chapter }{MappingGeneratorsImages}}{658}{subsection.40.10.2}%
\contentsline {subsection}{\numberline {40.10.3}\textcolor {Chapter }{IsGroupGeneralMappingByAsGroupGeneralMappingByImages}}{659}{subsection.40.10.3}%
\contentsline {subsection}{\numberline {40.10.4}\textcolor {Chapter }{IsPreimagesByAsGroupGeneralMappingByImages}}{659}{subsection.40.10.4}%
\contentsline {subsection}{\numberline {40.10.5}\textcolor {Chapter }{IsPermGroupGeneralMapping}}{659}{subsection.40.10.5}%
\contentsline {subsection}{\numberline {40.10.6}\textcolor {Chapter }{IsToPermGroupGeneralMappingByImages}}{659}{subsection.40.10.6}%
\contentsline {subsection}{\numberline {40.10.7}\textcolor {Chapter }{IsGroupGeneralMappingByPcgs}}{659}{subsection.40.10.7}%
\contentsline {subsection}{\numberline {40.10.8}\textcolor {Chapter }{IsPcGroupGeneralMappingByImages}}{659}{subsection.40.10.8}%
\contentsline {subsection}{\numberline {40.10.9}\textcolor {Chapter }{IsToPcGroupGeneralMappingByImages}}{659}{subsection.40.10.9}%
\contentsline {subsection}{\numberline {40.10.10}\textcolor {Chapter }{IsFromFpGroupGeneralMappingByImages}}{660}{subsection.40.10.10}%
\contentsline {subsection}{\numberline {40.10.11}\textcolor {Chapter }{IsFromFpGroupStdGensGeneralMappingByImages}}{660}{subsection.40.10.11}%
\contentsline {chapter}{\numberline {41}\textcolor {Chapter }{Group Actions}}{661}{chapter.41}%
\contentsline {section}{\numberline {41.1}\textcolor {Chapter }{About Group Actions}}{661}{section.41.1}%
\contentsline {section}{\numberline {41.2}\textcolor {Chapter }{Basic Actions}}{662}{section.41.2}%
\contentsline {subsection}{\numberline {41.2.1}\textcolor {Chapter }{OnPoints}}{662}{subsection.41.2.1}%
\contentsline {subsection}{\numberline {41.2.2}\textcolor {Chapter }{OnRight}}{663}{subsection.41.2.2}%
\contentsline {subsection}{\numberline {41.2.3}\textcolor {Chapter }{OnLeftInverse}}{663}{subsection.41.2.3}%
\contentsline {subsection}{\numberline {41.2.4}\textcolor {Chapter }{OnSets}}{663}{subsection.41.2.4}%
\contentsline {subsection}{\numberline {41.2.5}\textcolor {Chapter }{OnTuples}}{664}{subsection.41.2.5}%
\contentsline {subsection}{\numberline {41.2.6}\textcolor {Chapter }{OnPairs}}{664}{subsection.41.2.6}%
\contentsline {subsection}{\numberline {41.2.7}\textcolor {Chapter }{OnSetsSets}}{664}{subsection.41.2.7}%
\contentsline {subsection}{\numberline {41.2.8}\textcolor {Chapter }{OnSetsDisjointSets}}{665}{subsection.41.2.8}%
\contentsline {subsection}{\numberline {41.2.9}\textcolor {Chapter }{OnSetsTuples}}{665}{subsection.41.2.9}%
\contentsline {subsection}{\numberline {41.2.10}\textcolor {Chapter }{OnTuplesSets}}{665}{subsection.41.2.10}%
\contentsline {subsection}{\numberline {41.2.11}\textcolor {Chapter }{OnTuplesTuples}}{665}{subsection.41.2.11}%
\contentsline {subsection}{\numberline {41.2.12}\textcolor {Chapter }{OnLines}}{666}{subsection.41.2.12}%
\contentsline {subsection}{\numberline {41.2.13}\textcolor {Chapter }{OnIndeterminates (as a permutation action)}}{666}{subsection.41.2.13}%
\contentsline {subsection}{\numberline {41.2.14}\textcolor {Chapter }{Permuted (as a permutation action)}}{666}{subsection.41.2.14}%
\contentsline {subsection}{\numberline {41.2.15}\textcolor {Chapter }{OnSubspacesByCanonicalBasis}}{667}{subsection.41.2.15}%
\contentsline {section}{\numberline {41.3}\textcolor {Chapter }{Action on canonical representatives}}{667}{section.41.3}%
\contentsline {section}{\numberline {41.4}\textcolor {Chapter }{Orbits}}{667}{section.41.4}%
\contentsline {subsection}{\numberline {41.4.1}\textcolor {Chapter }{Orbit}}{668}{subsection.41.4.1}%
\contentsline {subsection}{\numberline {41.4.2}\textcolor {Chapter }{Orbits (operation)}}{668}{subsection.41.4.2}%
\contentsline {subsection}{\numberline {41.4.3}\textcolor {Chapter }{OrbitsDomain}}{668}{subsection.41.4.3}%
\contentsline {subsection}{\numberline {41.4.4}\textcolor {Chapter }{OrbitLength}}{669}{subsection.41.4.4}%
\contentsline {subsection}{\numberline {41.4.5}\textcolor {Chapter }{OrbitLengths}}{669}{subsection.41.4.5}%
\contentsline {subsection}{\numberline {41.4.6}\textcolor {Chapter }{OrbitLengthsDomain}}{669}{subsection.41.4.6}%
\contentsline {section}{\numberline {41.5}\textcolor {Chapter }{Stabilizers}}{670}{section.41.5}%
\contentsline {subsection}{\numberline {41.5.1}\textcolor {Chapter }{OrbitStabilizer}}{670}{subsection.41.5.1}%
\contentsline {subsection}{\numberline {41.5.2}\textcolor {Chapter }{Stabilizer}}{670}{subsection.41.5.2}%
\contentsline {subsection}{\numberline {41.5.3}\textcolor {Chapter }{OrbitStabilizerAlgorithm}}{671}{subsection.41.5.3}%
\contentsline {section}{\numberline {41.6}\textcolor {Chapter }{Elements with Prescribed Images}}{671}{section.41.6}%
\contentsline {subsection}{\numberline {41.6.1}\textcolor {Chapter }{RepresentativeAction}}{671}{subsection.41.6.1}%
\contentsline {section}{\numberline {41.7}\textcolor {Chapter }{The Permutation Image of an Action}}{672}{section.41.7}%
\contentsline {subsection}{\numberline {41.7.1}\textcolor {Chapter }{ActionHomomorphism}}{672}{subsection.41.7.1}%
\contentsline {subsection}{\numberline {41.7.2}\textcolor {Chapter }{Action (for a group, an action domain, etc.)}}{673}{subsection.41.7.2}%
\contentsline {subsection}{\numberline {41.7.3}\textcolor {Chapter }{SparseActionHomomorphism}}{673}{subsection.41.7.3}%
\contentsline {section}{\numberline {41.8}\textcolor {Chapter }{Action of a group on itself}}{674}{section.41.8}%
\contentsline {subsection}{\numberline {41.8.1}\textcolor {Chapter }{FactorCosetAction (for a group and subgroup)}}{674}{subsection.41.8.1}%
\contentsline {subsection}{\numberline {41.8.2}\textcolor {Chapter }{RegularActionHomomorphism}}{674}{subsection.41.8.2}%
\contentsline {subsection}{\numberline {41.8.3}\textcolor {Chapter }{AbelianSubfactorAction}}{675}{subsection.41.8.3}%
\contentsline {section}{\numberline {41.9}\textcolor {Chapter }{Permutations Induced by Elements and Cycles}}{675}{section.41.9}%
\contentsline {subsection}{\numberline {41.9.1}\textcolor {Chapter }{Permutation}}{675}{subsection.41.9.1}%
\contentsline {subsection}{\numberline {41.9.2}\textcolor {Chapter }{PermutationCycle}}{676}{subsection.41.9.2}%
\contentsline {subsection}{\numberline {41.9.3}\textcolor {Chapter }{Cycle}}{676}{subsection.41.9.3}%
\contentsline {subsection}{\numberline {41.9.4}\textcolor {Chapter }{CycleLength}}{676}{subsection.41.9.4}%
\contentsline {subsection}{\numberline {41.9.5}\textcolor {Chapter }{Cycles}}{676}{subsection.41.9.5}%
\contentsline {subsection}{\numberline {41.9.6}\textcolor {Chapter }{CycleLengths}}{676}{subsection.41.9.6}%
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\contentsline {section}{\numberline {41.10}\textcolor {Chapter }{Tests for Actions}}{677}{section.41.10}%
\contentsline {subsection}{\numberline {41.10.1}\textcolor {Chapter }{IsTransitive}}{677}{subsection.41.10.1}%
\contentsline {subsection}{\numberline {41.10.2}\textcolor {Chapter }{Transitivity}}{678}{subsection.41.10.2}%
\contentsline {subsection}{\numberline {41.10.3}\textcolor {Chapter }{RankAction}}{678}{subsection.41.10.3}%
\contentsline {subsection}{\numberline {41.10.4}\textcolor {Chapter }{IsSemiRegular}}{678}{subsection.41.10.4}%
\contentsline {subsection}{\numberline {41.10.5}\textcolor {Chapter }{IsRegular}}{679}{subsection.41.10.5}%
\contentsline {subsection}{\numberline {41.10.6}\textcolor {Chapter }{Earns}}{679}{subsection.41.10.6}%
\contentsline {subsection}{\numberline {41.10.7}\textcolor {Chapter }{IsPrimitive}}{679}{subsection.41.10.7}%
\contentsline {section}{\numberline {41.11}\textcolor {Chapter }{Block Systems}}{680}{section.41.11}%
\contentsline {subsection}{\numberline {41.11.1}\textcolor {Chapter }{Blocks}}{680}{subsection.41.11.1}%
\contentsline {subsection}{\numberline {41.11.2}\textcolor {Chapter }{MaximalBlocks}}{680}{subsection.41.11.2}%
\contentsline {subsection}{\numberline {41.11.3}\textcolor {Chapter }{RepresentativesMinimalBlocks}}{680}{subsection.41.11.3}%
\contentsline {subsection}{\numberline {41.11.4}\textcolor {Chapter }{AllBlocks}}{681}{subsection.41.11.4}%
\contentsline {section}{\numberline {41.12}\textcolor {Chapter }{External Sets}}{681}{section.41.12}%
\contentsline {subsection}{\numberline {41.12.1}\textcolor {Chapter }{IsExternalSet}}{682}{subsection.41.12.1}%
\contentsline {subsection}{\numberline {41.12.2}\textcolor {Chapter }{ExternalSet}}{682}{subsection.41.12.2}%
\contentsline {subsection}{\numberline {41.12.3}\textcolor {Chapter }{ActingDomain}}{682}{subsection.41.12.3}%
\contentsline {subsection}{\numberline {41.12.4}\textcolor {Chapter }{FunctionAction}}{682}{subsection.41.12.4}%
\contentsline {subsection}{\numberline {41.12.5}\textcolor {Chapter }{HomeEnumerator}}{683}{subsection.41.12.5}%
\contentsline {subsection}{\numberline {41.12.6}\textcolor {Chapter }{IsExternalSubset}}{683}{subsection.41.12.6}%
\contentsline {subsection}{\numberline {41.12.7}\textcolor {Chapter }{ExternalSubset}}{683}{subsection.41.12.7}%
\contentsline {subsection}{\numberline {41.12.8}\textcolor {Chapter }{IsExternalOrbit}}{683}{subsection.41.12.8}%
\contentsline {subsection}{\numberline {41.12.9}\textcolor {Chapter }{ExternalOrbit}}{683}{subsection.41.12.9}%
\contentsline {subsection}{\numberline {41.12.10}\textcolor {Chapter }{StabilizerOfExternalSet}}{684}{subsection.41.12.10}%
\contentsline {subsection}{\numberline {41.12.11}\textcolor {Chapter }{ExternalOrbits}}{684}{subsection.41.12.11}%
\contentsline {subsection}{\numberline {41.12.12}\textcolor {Chapter }{ExternalOrbitsStabilizers}}{684}{subsection.41.12.12}%
\contentsline {subsection}{\numberline {41.12.13}\textcolor {Chapter }{CanonicalRepresentativeOfExternalSet}}{684}{subsection.41.12.13}%
\contentsline {subsection}{\numberline {41.12.14}\textcolor {Chapter }{CanonicalRepresentativeDeterminatorOfExternalSet}}{685}{subsection.41.12.14}%
\contentsline {subsection}{\numberline {41.12.15}\textcolor {Chapter }{ActorOfExternalSet}}{685}{subsection.41.12.15}%
\contentsline {subsection}{\numberline {41.12.16}\textcolor {Chapter }{UnderlyingExternalSet}}{685}{subsection.41.12.16}%
\contentsline {subsection}{\numberline {41.12.17}\textcolor {Chapter }{SurjectiveActionHomomorphismAttr}}{685}{subsection.41.12.17}%
\contentsline {chapter}{\numberline {42}\textcolor {Chapter }{Permutations}}{687}{chapter.42}%
\contentsline {section}{\numberline {42.1}\textcolor {Chapter }{IsPerm (Filter)}}{687}{section.42.1}%
\contentsline {subsection}{\numberline {42.1.1}\textcolor {Chapter }{IsPerm}}{688}{subsection.42.1.1}%
\contentsline {subsection}{\numberline {42.1.2}\textcolor {Chapter }{IsPermCollection}}{688}{subsection.42.1.2}%
\contentsline {subsection}{\numberline {42.1.3}\textcolor {Chapter }{PermutationsFamily}}{688}{subsection.42.1.3}%
\contentsline {subsection}{\numberline {42.1.4}\textcolor {Chapter }{PERM{\textunderscore }INVERSE{\textunderscore }THRESHOLD}}{688}{subsection.42.1.4}%
\contentsline {section}{\numberline {42.2}\textcolor {Chapter }{Comparison of Permutations}}{689}{section.42.2}%
\contentsline {subsection}{\numberline {42.2.1}\textcolor {Chapter }{\texttt {\symbol {92}}= (for permutations)}}{689}{subsection.42.2.1}%
\contentsline {subsection}{\numberline {42.2.2}\textcolor {Chapter }{DistancePerms}}{689}{subsection.42.2.2}%
\contentsline {subsection}{\numberline {42.2.3}\textcolor {Chapter }{SmallestGeneratorPerm}}{689}{subsection.42.2.3}%
\contentsline {section}{\numberline {42.3}\textcolor {Chapter }{Moved Points of Permutations}}{689}{section.42.3}%
\contentsline {subsection}{\numberline {42.3.1}\textcolor {Chapter }{SmallestMovedPoint (for a permutation)}}{689}{subsection.42.3.1}%
\contentsline {subsection}{\numberline {42.3.2}\textcolor {Chapter }{LargestMovedPoint (for a permutation)}}{690}{subsection.42.3.2}%
\contentsline {subsection}{\numberline {42.3.3}\textcolor {Chapter }{MovedPoints (for a permutation)}}{690}{subsection.42.3.3}%
\contentsline {subsection}{\numberline {42.3.4}\textcolor {Chapter }{NrMovedPoints (for a permutation)}}{690}{subsection.42.3.4}%
\contentsline {section}{\numberline {42.4}\textcolor {Chapter }{Sign and Cycle Structure}}{691}{section.42.4}%
\contentsline {subsection}{\numberline {42.4.1}\textcolor {Chapter }{SignPerm}}{691}{subsection.42.4.1}%
\contentsline {subsection}{\numberline {42.4.2}\textcolor {Chapter }{CycleStructurePerm}}{691}{subsection.42.4.2}%
\contentsline {section}{\numberline {42.5}\textcolor {Chapter }{Creating Permutations}}{691}{section.42.5}%
\contentsline {subsection}{\numberline {42.5.1}\textcolor {Chapter }{ListPerm}}{691}{subsection.42.5.1}%
\contentsline {subsection}{\numberline {42.5.2}\textcolor {Chapter }{PermList}}{691}{subsection.42.5.2}%
\contentsline {subsection}{\numberline {42.5.3}\textcolor {Chapter }{MappingPermListList}}{692}{subsection.42.5.3}%
\contentsline {subsection}{\numberline {42.5.4}\textcolor {Chapter }{RestrictedPerm}}{692}{subsection.42.5.4}%
\contentsline {subsection}{\numberline {42.5.5}\textcolor {Chapter }{CycleFromList}}{692}{subsection.42.5.5}%
\contentsline {subsection}{\numberline {42.5.6}\textcolor {Chapter }{AsPermutation}}{693}{subsection.42.5.6}%
\contentsline {chapter}{\numberline {43}\textcolor {Chapter }{Permutation Groups}}{694}{chapter.43}%
\contentsline {section}{\numberline {43.1}\textcolor {Chapter }{IsPermGroup (Filter)}}{694}{section.43.1}%
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\contentsline {section}{\numberline {43.2}\textcolor {Chapter }{The Natural Action}}{694}{section.43.2}%
\contentsline {subsection}{\numberline {43.2.1}\textcolor {Chapter }{OrbitPerms}}{695}{subsection.43.2.1}%
\contentsline {subsection}{\numberline {43.2.2}\textcolor {Chapter }{OrbitsPerms}}{695}{subsection.43.2.2}%
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\contentsline {subsection}{\numberline {43.3.2}\textcolor {Chapter }{SmallerDegreePermutationRepresentation}}{695}{subsection.43.3.2}%
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\contentsline {subsection}{\numberline {43.4.2}\textcolor {Chapter }{IsSymmetricGroup}}{697}{subsection.43.4.2}%
\contentsline {subsection}{\numberline {43.4.3}\textcolor {Chapter }{IsAlternatingGroup}}{697}{subsection.43.4.3}%
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\contentsline {subsection}{\numberline {43.5.2}\textcolor {Chapter }{SocleTypePrimitiveGroup}}{698}{subsection.43.5.2}%
\contentsline {section}{\numberline {43.6}\textcolor {Chapter }{Stabilizer Chains}}{699}{section.43.6}%
\contentsline {section}{\numberline {43.7}\textcolor {Chapter }{Randomized Methods for Permutation Groups}}{699}{section.43.7}%
\contentsline {section}{\numberline {43.8}\textcolor {Chapter }{Construction of Stabilizer Chains}}{702}{section.43.8}%
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\contentsline {subsection}{\numberline {43.8.4}\textcolor {Chapter }{StabChainBaseStrongGenerators}}{704}{subsection.43.8.4}%
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\contentsline {section}{\numberline {43.10}\textcolor {Chapter }{Operations for Stabilizer Chains}}{705}{section.43.10}%
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\contentsline {subsection}{\numberline {43.10.5}\textcolor {Chapter }{GroupStabChain}}{706}{subsection.43.10.5}%
\contentsline {subsection}{\numberline {43.10.6}\textcolor {Chapter }{OrbitStabChain}}{706}{subsection.43.10.6}%
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\contentsline {subsection}{\numberline {43.10.9}\textcolor {Chapter }{ElementsStabChain}}{707}{subsection.43.10.9}%
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\contentsline {subsection}{\numberline {43.10.11}\textcolor {Chapter }{InverseRepresentative}}{707}{subsection.43.10.11}%
\contentsline {subsection}{\numberline {43.10.12}\textcolor {Chapter }{SiftedPermutation}}{707}{subsection.43.10.12}%
\contentsline {subsection}{\numberline {43.10.13}\textcolor {Chapter }{MinimalElementCosetStabChain}}{707}{subsection.43.10.13}%
\contentsline {subsection}{\numberline {43.10.14}\textcolor {Chapter }{LargestElementStabChain}}{707}{subsection.43.10.14}%
\contentsline {subsection}{\numberline {43.10.15}\textcolor {Chapter }{ApproximateSuborbitsStabilizerPermGroup}}{708}{subsection.43.10.15}%
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\contentsline {subsection}{\numberline {43.11.4}\textcolor {Chapter }{ExtendStabChain}}{709}{subsection.43.11.4}%
\contentsline {subsection}{\numberline {43.11.5}\textcolor {Chapter }{ReduceStabChain}}{709}{subsection.43.11.5}%
\contentsline {subsection}{\numberline {43.11.6}\textcolor {Chapter }{RemoveStabChain}}{709}{subsection.43.11.6}%
\contentsline {subsection}{\numberline {43.11.7}\textcolor {Chapter }{EmptyStabChain}}{709}{subsection.43.11.7}%
\contentsline {subsection}{\numberline {43.11.8}\textcolor {Chapter }{InsertTrivialStabilizer}}{709}{subsection.43.11.8}%
\contentsline {subsection}{\numberline {43.11.9}\textcolor {Chapter }{IsFixedStabilizer}}{709}{subsection.43.11.9}%
\contentsline {subsection}{\numberline {43.11.10}\textcolor {Chapter }{AddGeneratorsExtendSchreierTree}}{710}{subsection.43.11.10}%
\contentsline {section}{\numberline {43.12}\textcolor {Chapter }{Backtrack}}{710}{section.43.12}%
\contentsline {subsection}{\numberline {43.12.1}\textcolor {Chapter }{SubgroupProperty}}{710}{subsection.43.12.1}%
\contentsline {subsection}{\numberline {43.12.2}\textcolor {Chapter }{ElementProperty}}{710}{subsection.43.12.2}%
\contentsline {subsection}{\numberline {43.12.3}\textcolor {Chapter }{TwoClosure}}{711}{subsection.43.12.3}%
\contentsline {subsection}{\numberline {43.12.4}\textcolor {Chapter }{InfoBckt}}{711}{subsection.43.12.4}%
\contentsline {section}{\numberline {43.13}\textcolor {Chapter }{Working with large degree permutation groups}}{711}{section.43.13}%
\contentsline {chapter}{\numberline {44}\textcolor {Chapter }{Matrix Groups}}{713}{chapter.44}%
\contentsline {section}{\numberline {44.1}\textcolor {Chapter }{IsMatrixGroup (Filter)}}{713}{section.44.1}%
\contentsline {subsection}{\numberline {44.1.1}\textcolor {Chapter }{IsMatrixGroup}}{713}{subsection.44.1.1}%
\contentsline {section}{\numberline {44.2}\textcolor {Chapter }{Attributes and Properties for Matrix Groups}}{714}{section.44.2}%
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\contentsline {subsection}{\numberline {44.2.2}\textcolor {Chapter }{DefaultFieldOfMatrixGroup}}{714}{subsection.44.2.2}%
\contentsline {subsection}{\numberline {44.2.3}\textcolor {Chapter }{FieldOfMatrixGroup}}{714}{subsection.44.2.3}%
\contentsline {subsection}{\numberline {44.2.4}\textcolor {Chapter }{TransposedMatrixGroup}}{714}{subsection.44.2.4}%
\contentsline {subsection}{\numberline {44.2.5}\textcolor {Chapter }{IsFFEMatrixGroup}}{714}{subsection.44.2.5}%
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\contentsline {subsection}{\numberline {44.3.1}\textcolor {Chapter }{ProjectiveActionOnFullSpace}}{715}{subsection.44.3.1}%
\contentsline {subsection}{\numberline {44.3.2}\textcolor {Chapter }{ProjectiveActionHomomorphismMatrixGroup}}{715}{subsection.44.3.2}%
\contentsline {subsection}{\numberline {44.3.3}\textcolor {Chapter }{BlowUpIsomorphism}}{715}{subsection.44.3.3}%
\contentsline {section}{\numberline {44.4}\textcolor {Chapter }{GL and SL}}{715}{section.44.4}%
\contentsline {subsection}{\numberline {44.4.1}\textcolor {Chapter }{IsGeneralLinearGroup}}{716}{subsection.44.4.1}%
\contentsline {subsection}{\numberline {44.4.2}\textcolor {Chapter }{IsNaturalGL}}{716}{subsection.44.4.2}%
\contentsline {subsection}{\numberline {44.4.3}\textcolor {Chapter }{IsSpecialLinearGroup}}{716}{subsection.44.4.3}%
\contentsline {subsection}{\numberline {44.4.4}\textcolor {Chapter }{IsNaturalSL}}{716}{subsection.44.4.4}%
\contentsline {subsection}{\numberline {44.4.5}\textcolor {Chapter }{IsSubgroupSL}}{716}{subsection.44.4.5}%
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\contentsline {subsection}{\numberline {44.5.1}\textcolor {Chapter }{InvariantBilinearForm}}{717}{subsection.44.5.1}%
\contentsline {subsection}{\numberline {44.5.2}\textcolor {Chapter }{IsFullSubgroupGLorSLRespectingBilinearForm}}{717}{subsection.44.5.2}%
\contentsline {subsection}{\numberline {44.5.3}\textcolor {Chapter }{InvariantBilinearFormUpToScalars}}{717}{subsection.44.5.3}%
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\contentsline {subsection}{\numberline {44.5.5}\textcolor {Chapter }{InvariantSesquilinearForm}}{717}{subsection.44.5.5}%
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\contentsline {subsection}{\numberline {44.6.2}\textcolor {Chapter }{IsRationalMatrixGroup}}{719}{subsection.44.6.2}%
\contentsline {subsection}{\numberline {44.6.3}\textcolor {Chapter }{IsIntegerMatrixGroup}}{720}{subsection.44.6.3}%
\contentsline {subsection}{\numberline {44.6.4}\textcolor {Chapter }{IsNaturalGLnZ}}{720}{subsection.44.6.4}%
\contentsline {subsection}{\numberline {44.6.5}\textcolor {Chapter }{IsNaturalSLnZ}}{720}{subsection.44.6.5}%
\contentsline {subsection}{\numberline {44.6.6}\textcolor {Chapter }{InvariantLattice}}{720}{subsection.44.6.6}%
\contentsline {subsection}{\numberline {44.6.7}\textcolor {Chapter }{NormalizerInGLnZ}}{720}{subsection.44.6.7}%
\contentsline {subsection}{\numberline {44.6.8}\textcolor {Chapter }{CentralizerInGLnZ}}{721}{subsection.44.6.8}%
\contentsline {subsection}{\numberline {44.6.9}\textcolor {Chapter }{ZClassRepsQClass}}{721}{subsection.44.6.9}%
\contentsline {subsection}{\numberline {44.6.10}\textcolor {Chapter }{IsBravaisGroup}}{721}{subsection.44.6.10}%
\contentsline {subsection}{\numberline {44.6.11}\textcolor {Chapter }{BravaisGroup}}{721}{subsection.44.6.11}%
\contentsline {subsection}{\numberline {44.6.12}\textcolor {Chapter }{BravaisSubgroups}}{721}{subsection.44.6.12}%
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\contentsline {subsection}{\numberline {44.7.2}\textcolor {Chapter }{SetCrystGroupDefaultAction}}{722}{subsection.44.7.2}%
\contentsline {chapter}{\numberline {45}\textcolor {Chapter }{Polycyclic Groups}}{723}{chapter.45}%
\contentsline {section}{\numberline {45.1}\textcolor {Chapter }{Polycyclic Generating Systems}}{723}{section.45.1}%
\contentsline {section}{\numberline {45.2}\textcolor {Chapter }{Computing a Pcgs}}{724}{section.45.2}%
\contentsline {subsection}{\numberline {45.2.1}\textcolor {Chapter }{Pcgs}}{724}{subsection.45.2.1}%
\contentsline {subsection}{\numberline {45.2.2}\textcolor {Chapter }{IsPcgs}}{724}{subsection.45.2.2}%
\contentsline {subsection}{\numberline {45.2.3}\textcolor {Chapter }{CanEasilyComputePcgs}}{724}{subsection.45.2.3}%
\contentsline {section}{\numberline {45.3}\textcolor {Chapter }{Defining a Pcgs Yourself}}{725}{section.45.3}%
\contentsline {subsection}{\numberline {45.3.1}\textcolor {Chapter }{PcgsByPcSequence}}{725}{subsection.45.3.1}%
\contentsline {section}{\numberline {45.4}\textcolor {Chapter }{Elementary Operations for a Pcgs}}{725}{section.45.4}%
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\contentsline {subsection}{\numberline {45.4.2}\textcolor {Chapter }{IsFiniteOrdersPcgs}}{725}{subsection.45.4.2}%
\contentsline {subsection}{\numberline {45.4.3}\textcolor {Chapter }{IsPrimeOrdersPcgs}}{725}{subsection.45.4.3}%
\contentsline {subsection}{\numberline {45.4.4}\textcolor {Chapter }{PcSeries}}{726}{subsection.45.4.4}%
\contentsline {subsection}{\numberline {45.4.5}\textcolor {Chapter }{GroupOfPcgs}}{726}{subsection.45.4.5}%
\contentsline {subsection}{\numberline {45.4.6}\textcolor {Chapter }{OneOfPcgs}}{726}{subsection.45.4.6}%
\contentsline {section}{\numberline {45.5}\textcolor {Chapter }{Elementary Operations for a Pcgs and an Element}}{726}{section.45.5}%
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\contentsline {subsection}{\numberline {45.5.2}\textcolor {Chapter }{ExponentOfPcElement}}{726}{subsection.45.5.2}%
\contentsline {subsection}{\numberline {45.5.3}\textcolor {Chapter }{ExponentsOfPcElement}}{726}{subsection.45.5.3}%
\contentsline {subsection}{\numberline {45.5.4}\textcolor {Chapter }{DepthOfPcElement}}{727}{subsection.45.5.4}%
\contentsline {subsection}{\numberline {45.5.5}\textcolor {Chapter }{LeadingExponentOfPcElement}}{727}{subsection.45.5.5}%
\contentsline {subsection}{\numberline {45.5.6}\textcolor {Chapter }{PcElementByExponents}}{727}{subsection.45.5.6}%
\contentsline {subsection}{\numberline {45.5.7}\textcolor {Chapter }{LinearCombinationPcgs}}{727}{subsection.45.5.7}%
\contentsline {subsection}{\numberline {45.5.8}\textcolor {Chapter }{SiftedPcElement}}{727}{subsection.45.5.8}%
\contentsline {subsection}{\numberline {45.5.9}\textcolor {Chapter }{CanonicalPcElement}}{728}{subsection.45.5.9}%
\contentsline {subsection}{\numberline {45.5.10}\textcolor {Chapter }{ReducedPcElement}}{728}{subsection.45.5.10}%
\contentsline {subsection}{\numberline {45.5.11}\textcolor {Chapter }{CleanedTailPcElement}}{728}{subsection.45.5.11}%
\contentsline {subsection}{\numberline {45.5.12}\textcolor {Chapter }{HeadPcElementByNumber}}{728}{subsection.45.5.12}%
\contentsline {section}{\numberline {45.6}\textcolor {Chapter }{Exponents of Special Products}}{728}{section.45.6}%
\contentsline {subsection}{\numberline {45.6.1}\textcolor {Chapter }{ExponentsConjugateLayer}}{729}{subsection.45.6.1}%
\contentsline {subsection}{\numberline {45.6.2}\textcolor {Chapter }{ExponentsOfRelativePower}}{729}{subsection.45.6.2}%
\contentsline {subsection}{\numberline {45.6.3}\textcolor {Chapter }{ExponentsOfConjugate}}{729}{subsection.45.6.3}%
\contentsline {subsection}{\numberline {45.6.4}\textcolor {Chapter }{ExponentsOfCommutator}}{729}{subsection.45.6.4}%
\contentsline {section}{\numberline {45.7}\textcolor {Chapter }{Subgroups of Polycyclic Groups {\textendash } Induced Pcgs}}{729}{section.45.7}%
\contentsline {subsection}{\numberline {45.7.1}\textcolor {Chapter }{IsInducedPcgs}}{730}{subsection.45.7.1}%
\contentsline {subsection}{\numberline {45.7.2}\textcolor {Chapter }{InducedPcgsByPcSequence}}{730}{subsection.45.7.2}%
\contentsline {subsection}{\numberline {45.7.3}\textcolor {Chapter }{ParentPcgs}}{730}{subsection.45.7.3}%
\contentsline {subsection}{\numberline {45.7.4}\textcolor {Chapter }{InducedPcgs}}{730}{subsection.45.7.4}%
\contentsline {subsection}{\numberline {45.7.5}\textcolor {Chapter }{InducedPcgsByGenerators}}{731}{subsection.45.7.5}%
\contentsline {subsection}{\numberline {45.7.6}\textcolor {Chapter }{InducedPcgsByPcSequenceAndGenerators}}{731}{subsection.45.7.6}%
\contentsline {subsection}{\numberline {45.7.7}\textcolor {Chapter }{LeadCoeffsIGS}}{731}{subsection.45.7.7}%
\contentsline {subsection}{\numberline {45.7.8}\textcolor {Chapter }{ExtendedPcgs}}{731}{subsection.45.7.8}%
\contentsline {subsection}{\numberline {45.7.9}\textcolor {Chapter }{SubgroupByPcgs}}{731}{subsection.45.7.9}%
\contentsline {section}{\numberline {45.8}\textcolor {Chapter }{Subgroups of Polycyclic Groups {\textendash } Canonical Pcgs}}{731}{section.45.8}%
\contentsline {subsection}{\numberline {45.8.1}\textcolor {Chapter }{IsCanonicalPcgs}}{732}{subsection.45.8.1}%
\contentsline {subsection}{\numberline {45.8.2}\textcolor {Chapter }{CanonicalPcgs}}{732}{subsection.45.8.2}%
\contentsline {section}{\numberline {45.9}\textcolor {Chapter }{Factor Groups of Polycyclic Groups {\textendash } Modulo Pcgs}}{732}{section.45.9}%
\contentsline {subsection}{\numberline {45.9.1}\textcolor {Chapter }{ModuloPcgs}}{732}{subsection.45.9.1}%
\contentsline {subsection}{\numberline {45.9.2}\textcolor {Chapter }{IsModuloPcgs}}{733}{subsection.45.9.2}%
\contentsline {subsection}{\numberline {45.9.3}\textcolor {Chapter }{NumeratorOfModuloPcgs}}{733}{subsection.45.9.3}%
\contentsline {subsection}{\numberline {45.9.4}\textcolor {Chapter }{DenominatorOfModuloPcgs}}{733}{subsection.45.9.4}%
\contentsline {subsection}{\numberline {45.9.5}\textcolor {Chapter }{\texttt {\symbol {92}}mod (for two pcgs)}}{733}{subsection.45.9.5}%
\contentsline {subsection}{\numberline {45.9.6}\textcolor {Chapter }{CorrespondingGeneratorsByModuloPcgs}}{734}{subsection.45.9.6}%
\contentsline {subsection}{\numberline {45.9.7}\textcolor {Chapter }{CanonicalPcgsByGeneratorsWithImages}}{734}{subsection.45.9.7}%
\contentsline {section}{\numberline {45.10}\textcolor {Chapter }{Factor Groups of Polycyclic Groups in their Own Representation}}{734}{section.45.10}%
\contentsline {subsection}{\numberline {45.10.1}\textcolor {Chapter }{ProjectedPcElement}}{734}{subsection.45.10.1}%
\contentsline {subsection}{\numberline {45.10.2}\textcolor {Chapter }{ProjectedInducedPcgs}}{734}{subsection.45.10.2}%
\contentsline {subsection}{\numberline {45.10.3}\textcolor {Chapter }{LiftedPcElement}}{735}{subsection.45.10.3}%
\contentsline {subsection}{\numberline {45.10.4}\textcolor {Chapter }{LiftedInducedPcgs}}{735}{subsection.45.10.4}%
\contentsline {section}{\numberline {45.11}\textcolor {Chapter }{Pcgs and Normal Series}}{735}{section.45.11}%
\contentsline {subsection}{\numberline {45.11.1}\textcolor {Chapter }{IsPcgsElementaryAbelianSeries}}{735}{subsection.45.11.1}%
\contentsline {subsection}{\numberline {45.11.2}\textcolor {Chapter }{PcgsElementaryAbelianSeries (for a group)}}{735}{subsection.45.11.2}%
\contentsline {subsection}{\numberline {45.11.3}\textcolor {Chapter }{IndicesEANormalSteps}}{735}{subsection.45.11.3}%
\contentsline {subsection}{\numberline {45.11.4}\textcolor {Chapter }{EANormalSeriesByPcgs}}{736}{subsection.45.11.4}%
\contentsline {subsection}{\numberline {45.11.5}\textcolor {Chapter }{IsPcgsCentralSeries}}{736}{subsection.45.11.5}%
\contentsline {subsection}{\numberline {45.11.6}\textcolor {Chapter }{PcgsCentralSeries}}{736}{subsection.45.11.6}%
\contentsline {subsection}{\numberline {45.11.7}\textcolor {Chapter }{IndicesCentralNormalSteps}}{736}{subsection.45.11.7}%
\contentsline {subsection}{\numberline {45.11.8}\textcolor {Chapter }{CentralNormalSeriesByPcgs}}{737}{subsection.45.11.8}%
\contentsline {subsection}{\numberline {45.11.9}\textcolor {Chapter }{IsPcgsPCentralSeriesPGroup}}{737}{subsection.45.11.9}%
\contentsline {subsection}{\numberline {45.11.10}\textcolor {Chapter }{PcgsPCentralSeriesPGroup}}{737}{subsection.45.11.10}%
\contentsline {subsection}{\numberline {45.11.11}\textcolor {Chapter }{IndicesPCentralNormalStepsPGroup}}{737}{subsection.45.11.11}%
\contentsline {subsection}{\numberline {45.11.12}\textcolor {Chapter }{PCentralNormalSeriesByPcgsPGroup}}{738}{subsection.45.11.12}%
\contentsline {subsection}{\numberline {45.11.13}\textcolor {Chapter }{IsPcgsChiefSeries}}{738}{subsection.45.11.13}%
\contentsline {subsection}{\numberline {45.11.14}\textcolor {Chapter }{PcgsChiefSeries}}{738}{subsection.45.11.14}%
\contentsline {subsection}{\numberline {45.11.15}\textcolor {Chapter }{IndicesChiefNormalSteps}}{738}{subsection.45.11.15}%
\contentsline {subsection}{\numberline {45.11.16}\textcolor {Chapter }{ChiefNormalSeriesByPcgs}}{738}{subsection.45.11.16}%
\contentsline {subsection}{\numberline {45.11.17}\textcolor {Chapter }{IndicesNormalSteps}}{739}{subsection.45.11.17}%
\contentsline {subsection}{\numberline {45.11.18}\textcolor {Chapter }{NormalSeriesByPcgs}}{739}{subsection.45.11.18}%
\contentsline {section}{\numberline {45.12}\textcolor {Chapter }{Sum and Intersection of Pcgs}}{739}{section.45.12}%
\contentsline {subsection}{\numberline {45.12.1}\textcolor {Chapter }{SumFactorizationFunctionPcgs}}{739}{subsection.45.12.1}%
\contentsline {section}{\numberline {45.13}\textcolor {Chapter }{Special Pcgs}}{740}{section.45.13}%
\contentsline {subsection}{\numberline {45.13.1}\textcolor {Chapter }{IsSpecialPcgs}}{741}{subsection.45.13.1}%
\contentsline {subsection}{\numberline {45.13.2}\textcolor {Chapter }{SpecialPcgs}}{741}{subsection.45.13.2}%
\contentsline {subsection}{\numberline {45.13.3}\textcolor {Chapter }{LGWeights}}{741}{subsection.45.13.3}%
\contentsline {subsection}{\numberline {45.13.4}\textcolor {Chapter }{LGLayers}}{741}{subsection.45.13.4}%
\contentsline {subsection}{\numberline {45.13.5}\textcolor {Chapter }{LGFirst}}{741}{subsection.45.13.5}%
\contentsline {subsection}{\numberline {45.13.6}\textcolor {Chapter }{LGLength}}{741}{subsection.45.13.6}%
\contentsline {subsection}{\numberline {45.13.7}\textcolor {Chapter }{IsInducedPcgsWrtSpecialPcgs}}{742}{subsection.45.13.7}%
\contentsline {subsection}{\numberline {45.13.8}\textcolor {Chapter }{InducedPcgsWrtSpecialPcgs}}{742}{subsection.45.13.8}%
\contentsline {section}{\numberline {45.14}\textcolor {Chapter }{Action on Subfactors Defined by a Pcgs}}{742}{section.45.14}%
\contentsline {subsection}{\numberline {45.14.1}\textcolor {Chapter }{VectorSpaceByPcgsOfElementaryAbelianGroup}}{742}{subsection.45.14.1}%
\contentsline {subsection}{\numberline {45.14.2}\textcolor {Chapter }{LinearAction}}{743}{subsection.45.14.2}%
\contentsline {subsection}{\numberline {45.14.3}\textcolor {Chapter }{LinearActionLayer}}{743}{subsection.45.14.3}%
\contentsline {subsection}{\numberline {45.14.4}\textcolor {Chapter }{AffineAction}}{743}{subsection.45.14.4}%
\contentsline {subsection}{\numberline {45.14.5}\textcolor {Chapter }{AffineActionLayer}}{743}{subsection.45.14.5}%
\contentsline {section}{\numberline {45.15}\textcolor {Chapter }{Orbit Stabilizer Methods for Polycyclic Groups}}{744}{section.45.15}%
\contentsline {subsection}{\numberline {45.15.1}\textcolor {Chapter }{StabilizerPcgs}}{744}{subsection.45.15.1}%
\contentsline {subsection}{\numberline {45.15.2}\textcolor {Chapter }{Pcgs{\textunderscore }OrbitStabilizer}}{744}{subsection.45.15.2}%
\contentsline {section}{\numberline {45.16}\textcolor {Chapter }{Operations which have Special Methods for Groups with Pcgs}}{744}{section.45.16}%
\contentsline {section}{\numberline {45.17}\textcolor {Chapter }{Conjugacy Classes in Solvable Groups}}{744}{section.45.17}%
\contentsline {subsection}{\numberline {45.17.1}\textcolor {Chapter }{ClassesSolvableGroup}}{745}{subsection.45.17.1}%
\contentsline {subsection}{\numberline {45.17.2}\textcolor {Chapter }{CentralizerSizeLimitConsiderFunction}}{745}{subsection.45.17.2}%
\contentsline {chapter}{\numberline {46}\textcolor {Chapter }{Pc Groups}}{746}{chapter.46}%
\contentsline {section}{\numberline {46.1}\textcolor {Chapter }{The Family Pcgs}}{747}{section.46.1}%
\contentsline {subsection}{\numberline {46.1.1}\textcolor {Chapter }{FamilyPcgs}}{747}{subsection.46.1.1}%
\contentsline {subsection}{\numberline {46.1.2}\textcolor {Chapter }{IsFamilyPcgs}}{747}{subsection.46.1.2}%
\contentsline {subsection}{\numberline {46.1.3}\textcolor {Chapter }{InducedPcgsWrtFamilyPcgs}}{747}{subsection.46.1.3}%
\contentsline {subsection}{\numberline {46.1.4}\textcolor {Chapter }{IsParentPcgsFamilyPcgs}}{747}{subsection.46.1.4}%
\contentsline {section}{\numberline {46.2}\textcolor {Chapter }{Elements of Pc Groups}}{748}{section.46.2}%
\contentsline {subsection}{\numberline {46.2.1}\textcolor {Chapter }{Comparison of elements of pc groups}}{748}{subsection.46.2.1}%
\contentsline {subsection}{\numberline {46.2.2}\textcolor {Chapter }{Arithmetic operations for elements of pc groups}}{748}{subsection.46.2.2}%
\contentsline {section}{\numberline {46.3}\textcolor {Chapter }{Pc Groups versus Fp Groups}}{748}{section.46.3}%
\contentsline {subsection}{\numberline {46.3.1}\textcolor {Chapter }{IsPcGroup}}{748}{subsection.46.3.1}%
\contentsline {subsection}{\numberline {46.3.2}\textcolor {Chapter }{IsomorphismFpGroupByPcgs}}{749}{subsection.46.3.2}%
\contentsline {section}{\numberline {46.4}\textcolor {Chapter }{Constructing Pc Groups}}{749}{section.46.4}%
\contentsline {subsection}{\numberline {46.4.1}\textcolor {Chapter }{PcGroupFpGroup}}{750}{subsection.46.4.1}%
\contentsline {subsection}{\numberline {46.4.2}\textcolor {Chapter }{SingleCollector}}{750}{subsection.46.4.2}%
\contentsline {subsection}{\numberline {46.4.3}\textcolor {Chapter }{SetConjugate}}{750}{subsection.46.4.3}%
\contentsline {subsection}{\numberline {46.4.4}\textcolor {Chapter }{SetCommutator}}{751}{subsection.46.4.4}%
\contentsline {subsection}{\numberline {46.4.5}\textcolor {Chapter }{SetPower}}{751}{subsection.46.4.5}%
\contentsline {subsection}{\numberline {46.4.6}\textcolor {Chapter }{GroupByRws}}{751}{subsection.46.4.6}%
\contentsline {subsection}{\numberline {46.4.7}\textcolor {Chapter }{IsConfluent (for pc groups)}}{751}{subsection.46.4.7}%
\contentsline {subsection}{\numberline {46.4.8}\textcolor {Chapter }{IsomorphismRefinedPcGroup}}{752}{subsection.46.4.8}%
\contentsline {subsection}{\numberline {46.4.9}\textcolor {Chapter }{RefinedPcGroup}}{752}{subsection.46.4.9}%
\contentsline {section}{\numberline {46.5}\textcolor {Chapter }{Computing Pc Groups}}{752}{section.46.5}%
\contentsline {subsection}{\numberline {46.5.1}\textcolor {Chapter }{PcGroupWithPcgs}}{752}{subsection.46.5.1}%
\contentsline {subsection}{\numberline {46.5.2}\textcolor {Chapter }{IsomorphismPcGroup}}{753}{subsection.46.5.2}%
\contentsline {subsection}{\numberline {46.5.3}\textcolor {Chapter }{IsomorphismSpecialPcGroup}}{753}{subsection.46.5.3}%
\contentsline {section}{\numberline {46.6}\textcolor {Chapter }{Saving a Pc Group}}{753}{section.46.6}%
\contentsline {subsection}{\numberline {46.6.1}\textcolor {Chapter }{GapInputPcGroup}}{753}{subsection.46.6.1}%
\contentsline {section}{\numberline {46.7}\textcolor {Chapter }{Operations for Pc Groups}}{753}{section.46.7}%
\contentsline {section}{\numberline {46.8}\textcolor {Chapter }{$2$\texttt {\symbol {45}}Cohomology and Extensions}}{754}{section.46.8}%
\contentsline {subsection}{\numberline {46.8.1}\textcolor {Chapter }{TwoCoboundaries}}{754}{subsection.46.8.1}%
\contentsline {subsection}{\numberline {46.8.2}\textcolor {Chapter }{TwoCocycles}}{754}{subsection.46.8.2}%
\contentsline {subsection}{\numberline {46.8.3}\textcolor {Chapter }{TwoCohomology}}{754}{subsection.46.8.3}%
\contentsline {subsection}{\numberline {46.8.4}\textcolor {Chapter }{Extensions}}{755}{subsection.46.8.4}%
\contentsline {subsection}{\numberline {46.8.5}\textcolor {Chapter }{Extension}}{755}{subsection.46.8.5}%
\contentsline {subsection}{\numberline {46.8.6}\textcolor {Chapter }{SplitExtension}}{755}{subsection.46.8.6}%
\contentsline {subsection}{\numberline {46.8.7}\textcolor {Chapter }{ModuleOfExtension}}{755}{subsection.46.8.7}%
\contentsline {subsection}{\numberline {46.8.8}\textcolor {Chapter }{CompatiblePairs}}{756}{subsection.46.8.8}%
\contentsline {subsection}{\numberline {46.8.9}\textcolor {Chapter }{ExtensionRepresentatives}}{756}{subsection.46.8.9}%
\contentsline {subsection}{\numberline {46.8.10}\textcolor {Chapter }{SplitExtension (with specified homomorphism)}}{757}{subsection.46.8.10}%
\contentsline {section}{\numberline {46.9}\textcolor {Chapter }{Coding a Pc Presentation}}{758}{section.46.9}%
\contentsline {subsection}{\numberline {46.9.1}\textcolor {Chapter }{CodePcgs}}{758}{subsection.46.9.1}%
\contentsline {subsection}{\numberline {46.9.2}\textcolor {Chapter }{CodePcGroup}}{758}{subsection.46.9.2}%
\contentsline {subsection}{\numberline {46.9.3}\textcolor {Chapter }{PcGroupCode}}{758}{subsection.46.9.3}%
\contentsline {section}{\numberline {46.10}\textcolor {Chapter }{Random Isomorphism Testing}}{759}{section.46.10}%
\contentsline {subsection}{\numberline {46.10.1}\textcolor {Chapter }{RandomIsomorphismTest}}{759}{subsection.46.10.1}%
\contentsline {chapter}{\numberline {47}\textcolor {Chapter }{Finitely Presented Groups}}{760}{chapter.47}%
\contentsline {section}{\numberline {47.1}\textcolor {Chapter }{IsSubgroupFpGroup and IsFpGroup}}{761}{section.47.1}%
\contentsline {subsection}{\numberline {47.1.1}\textcolor {Chapter }{IsSubgroupFpGroup}}{761}{subsection.47.1.1}%
\contentsline {subsection}{\numberline {47.1.2}\textcolor {Chapter }{IsFpGroup}}{762}{subsection.47.1.2}%
\contentsline {subsection}{\numberline {47.1.3}\textcolor {Chapter }{InfoFpGroup}}{762}{subsection.47.1.3}%
\contentsline {section}{\numberline {47.2}\textcolor {Chapter }{Creating Finitely Presented Groups}}{762}{section.47.2}%
\contentsline {subsection}{\numberline {47.2.1}\textcolor {Chapter }{\texttt {\symbol {92}}/ (for a free group and a list of elements)}}{762}{subsection.47.2.1}%
\contentsline {subsection}{\numberline {47.2.2}\textcolor {Chapter }{FactorGroupFpGroupByRels}}{763}{subsection.47.2.2}%
\contentsline {subsection}{\numberline {47.2.3}\textcolor {Chapter }{ParseRelators}}{763}{subsection.47.2.3}%
\contentsline {subsection}{\numberline {47.2.4}\textcolor {Chapter }{StringFactorizationWord}}{763}{subsection.47.2.4}%
\contentsline {section}{\numberline {47.3}\textcolor {Chapter }{Comparison of Elements of Finitely Presented Groups}}{763}{section.47.3}%
\contentsline {subsection}{\numberline {47.3.1}\textcolor {Chapter }{\texttt {\symbol {92}}= (for two elements in a f.p. group)}}{763}{subsection.47.3.1}%
\contentsline {subsection}{\numberline {47.3.2}\textcolor {Chapter }{\texttt {\symbol {92}}{\textless } (for two elements in a f.p. group)}}{764}{subsection.47.3.2}%
\contentsline {subsection}{\numberline {47.3.3}\textcolor {Chapter }{FpElmComparisonMethod}}{764}{subsection.47.3.3}%
\contentsline {subsection}{\numberline {47.3.4}\textcolor {Chapter }{SetReducedMultiplication}}{764}{subsection.47.3.4}%
\contentsline {section}{\numberline {47.4}\textcolor {Chapter }{Preimages in the Free Group}}{764}{section.47.4}%
\contentsline {subsection}{\numberline {47.4.1}\textcolor {Chapter }{FreeGroupOfFpGroup}}{764}{subsection.47.4.1}%
\contentsline {subsection}{\numberline {47.4.2}\textcolor {Chapter }{FreeGeneratorsOfFpGroup}}{765}{subsection.47.4.2}%
\contentsline {subsection}{\numberline {47.4.3}\textcolor {Chapter }{RelatorsOfFpGroup}}{765}{subsection.47.4.3}%
\contentsline {subsection}{\numberline {47.4.4}\textcolor {Chapter }{UnderlyingElement (fp group elements)}}{765}{subsection.47.4.4}%
\contentsline {subsection}{\numberline {47.4.5}\textcolor {Chapter }{ElementOfFpGroup}}{766}{subsection.47.4.5}%
\contentsline {section}{\numberline {47.5}\textcolor {Chapter }{Operations for Finitely Presented Groups}}{766}{section.47.5}%
\contentsline {subsection}{\numberline {47.5.1}\textcolor {Chapter }{PseudoRandom (for finitely presented groups)}}{766}{subsection.47.5.1}%
\contentsline {section}{\numberline {47.6}\textcolor {Chapter }{Coset Tables and Coset Enumeration}}{767}{section.47.6}%
\contentsline {subsection}{\numberline {47.6.1}\textcolor {Chapter }{CosetTable}}{767}{subsection.47.6.1}%
\contentsline {subsection}{\numberline {47.6.2}\textcolor {Chapter }{TracedCosetFpGroup}}{768}{subsection.47.6.2}%
\contentsline {subsection}{\numberline {47.6.3}\textcolor {Chapter }{FactorCosetAction (for fp groups)}}{768}{subsection.47.6.3}%
\contentsline {subsection}{\numberline {47.6.4}\textcolor {Chapter }{CosetTableBySubgroup}}{768}{subsection.47.6.4}%
\contentsline {subsection}{\numberline {47.6.5}\textcolor {Chapter }{CosetTableFromGensAndRels}}{768}{subsection.47.6.5}%
\contentsline {subsection}{\numberline {47.6.6}\textcolor {Chapter }{CosetTableDefaultMaxLimit}}{769}{subsection.47.6.6}%
\contentsline {subsection}{\numberline {47.6.7}\textcolor {Chapter }{CosetTableDefaultLimit}}{770}{subsection.47.6.7}%
\contentsline {subsection}{\numberline {47.6.8}\textcolor {Chapter }{MostFrequentGeneratorFpGroup}}{770}{subsection.47.6.8}%
\contentsline {subsection}{\numberline {47.6.9}\textcolor {Chapter }{IndicesInvolutaryGenerators}}{771}{subsection.47.6.9}%
\contentsline {section}{\numberline {47.7}\textcolor {Chapter }{Standardization of coset tables}}{771}{section.47.7}%
\contentsline {subsection}{\numberline {47.7.1}\textcolor {Chapter }{CosetTableStandard}}{771}{subsection.47.7.1}%
\contentsline {subsection}{\numberline {47.7.2}\textcolor {Chapter }{StandardizeTable}}{771}{subsection.47.7.2}%
\contentsline {section}{\numberline {47.8}\textcolor {Chapter }{Coset tables for subgroups in the whole group}}{772}{section.47.8}%
\contentsline {subsection}{\numberline {47.8.1}\textcolor {Chapter }{CosetTableInWholeGroup}}{772}{subsection.47.8.1}%
\contentsline {subsection}{\numberline {47.8.2}\textcolor {Chapter }{SubgroupOfWholeGroupByCosetTable}}{772}{subsection.47.8.2}%
\contentsline {section}{\numberline {47.9}\textcolor {Chapter }{Augmented Coset Tables and Rewriting}}{772}{section.47.9}%
\contentsline {subsection}{\numberline {47.9.1}\textcolor {Chapter }{AugmentedCosetTableInWholeGroup}}{772}{subsection.47.9.1}%
\contentsline {subsection}{\numberline {47.9.2}\textcolor {Chapter }{AugmentedCosetTableMtc}}{773}{subsection.47.9.2}%
\contentsline {subsection}{\numberline {47.9.3}\textcolor {Chapter }{AugmentedCosetTableRrs}}{773}{subsection.47.9.3}%
\contentsline {subsection}{\numberline {47.9.4}\textcolor {Chapter }{RewriteWord}}{773}{subsection.47.9.4}%
\contentsline {section}{\numberline {47.10}\textcolor {Chapter }{Low Index Subgroups}}{773}{section.47.10}%
\contentsline {subsection}{\numberline {47.10.1}\textcolor {Chapter }{LowIndexSubgroupsFpGroupIterator}}{773}{subsection.47.10.1}%
\contentsline {section}{\numberline {47.11}\textcolor {Chapter }{Converting Groups to Finitely Presented Groups}}{775}{section.47.11}%
\contentsline {subsection}{\numberline {47.11.1}\textcolor {Chapter }{IsomorphismFpGroup}}{775}{subsection.47.11.1}%
\contentsline {subsection}{\numberline {47.11.2}\textcolor {Chapter }{IsomorphismFpGroupByGenerators}}{775}{subsection.47.11.2}%
\contentsline {section}{\numberline {47.12}\textcolor {Chapter }{New Presentations and Presentations for Subgroups}}{777}{section.47.12}%
\contentsline {subsection}{\numberline {47.12.1}\textcolor {Chapter }{IsomorphismSimplifiedFpGroup}}{778}{subsection.47.12.1}%
\contentsline {section}{\numberline {47.13}\textcolor {Chapter }{Preimages under Homomorphisms from an FpGroup}}{778}{section.47.13}%
\contentsline {subsection}{\numberline {47.13.1}\textcolor {Chapter }{SubgroupOfWholeGroupByQuotientSubgroup}}{779}{subsection.47.13.1}%
\contentsline {subsection}{\numberline {47.13.2}\textcolor {Chapter }{IsSubgroupOfWholeGroupByQuotientRep}}{779}{subsection.47.13.2}%
\contentsline {subsection}{\numberline {47.13.3}\textcolor {Chapter }{AsSubgroupOfWholeGroupByQuotient}}{779}{subsection.47.13.3}%
\contentsline {subsection}{\numberline {47.13.4}\textcolor {Chapter }{DefiningQuotientHomomorphism}}{779}{subsection.47.13.4}%
\contentsline {section}{\numberline {47.14}\textcolor {Chapter }{Quotient Methods}}{779}{section.47.14}%
\contentsline {subsection}{\numberline {47.14.1}\textcolor {Chapter }{PQuotient}}{780}{subsection.47.14.1}%
\contentsline {subsection}{\numberline {47.14.2}\textcolor {Chapter }{EpimorphismQuotientSystem}}{780}{subsection.47.14.2}%
\contentsline {subsection}{\numberline {47.14.3}\textcolor {Chapter }{EpimorphismPGroup}}{781}{subsection.47.14.3}%
\contentsline {subsection}{\numberline {47.14.4}\textcolor {Chapter }{EpimorphismNilpotentQuotient}}{781}{subsection.47.14.4}%
\contentsline {subsection}{\numberline {47.14.5}\textcolor {Chapter }{SolvableQuotient (for a f.p. group and a size)}}{781}{subsection.47.14.5}%
\contentsline {subsection}{\numberline {47.14.6}\textcolor {Chapter }{EpimorphismSolvableQuotient}}{782}{subsection.47.14.6}%
\contentsline {subsection}{\numberline {47.14.7}\textcolor {Chapter }{LargerQuotientBySubgroupAbelianization}}{782}{subsection.47.14.7}%
\contentsline {section}{\numberline {47.15}\textcolor {Chapter }{Abelian Invariants for Subgroups}}{783}{section.47.15}%
\contentsline {subsection}{\numberline {47.15.1}\textcolor {Chapter }{AbelianInvariantsSubgroupFpGroup}}{783}{subsection.47.15.1}%
\contentsline {subsection}{\numberline {47.15.2}\textcolor {Chapter }{AbelianInvariantsSubgroupFpGroupMtc}}{783}{subsection.47.15.2}%
\contentsline {subsection}{\numberline {47.15.3}\textcolor {Chapter }{AbelianInvariantsSubgroupFpGroupRrs}}{783}{subsection.47.15.3}%
\contentsline {subsection}{\numberline {47.15.4}\textcolor {Chapter }{AbelianInvariantsNormalClosureFpGroup}}{783}{subsection.47.15.4}%
\contentsline {subsection}{\numberline {47.15.5}\textcolor {Chapter }{AbelianInvariantsNormalClosureFpGroupRrs}}{783}{subsection.47.15.5}%
\contentsline {section}{\numberline {47.16}\textcolor {Chapter }{Testing Finiteness of Finitely Presented Groups}}{784}{section.47.16}%
\contentsline {subsection}{\numberline {47.16.1}\textcolor {Chapter }{IsInfiniteAbelianizationGroup}}{785}{subsection.47.16.1}%
\contentsline {subsection}{\numberline {47.16.2}\textcolor {Chapter }{NewmanInfinityCriterion}}{785}{subsection.47.16.2}%
\contentsline {chapter}{\numberline {48}\textcolor {Chapter }{Presentations and Tietze Transformations}}{786}{chapter.48}%
\contentsline {section}{\numberline {48.1}\textcolor {Chapter }{Creating Presentations}}{786}{section.48.1}%
\contentsline {subsection}{\numberline {48.1.1}\textcolor {Chapter }{PresentationFpGroup}}{786}{subsection.48.1.1}%
\contentsline {subsection}{\numberline {48.1.2}\textcolor {Chapter }{TzSort}}{787}{subsection.48.1.2}%
\contentsline {subsection}{\numberline {48.1.3}\textcolor {Chapter }{GeneratorsOfPresentation}}{787}{subsection.48.1.3}%
\contentsline {subsection}{\numberline {48.1.4}\textcolor {Chapter }{FpGroupPresentation}}{787}{subsection.48.1.4}%
\contentsline {subsection}{\numberline {48.1.5}\textcolor {Chapter }{PresentationViaCosetTable}}{787}{subsection.48.1.5}%
\contentsline {subsection}{\numberline {48.1.6}\textcolor {Chapter }{SimplifiedFpGroup}}{789}{subsection.48.1.6}%
\contentsline {section}{\numberline {48.2}\textcolor {Chapter }{Subgroup Presentations}}{789}{section.48.2}%
\contentsline {subsection}{\numberline {48.2.1}\textcolor {Chapter }{PresentationSubgroup}}{789}{subsection.48.2.1}%
\contentsline {subsection}{\numberline {48.2.2}\textcolor {Chapter }{PresentationSubgroupRrs}}{789}{subsection.48.2.2}%
\contentsline {subsection}{\numberline {48.2.3}\textcolor {Chapter }{PrimaryGeneratorWords}}{791}{subsection.48.2.3}%
\contentsline {subsection}{\numberline {48.2.4}\textcolor {Chapter }{PresentationSubgroupMtc}}{791}{subsection.48.2.4}%
\contentsline {subsection}{\numberline {48.2.5}\textcolor {Chapter }{PresentationNormalClosureRrs}}{793}{subsection.48.2.5}%
\contentsline {subsection}{\numberline {48.2.6}\textcolor {Chapter }{PresentationNormalClosure}}{793}{subsection.48.2.6}%
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\contentsline {section}{\numberline {48.11}\textcolor {Chapter }{Tietze Options}}{810}{section.48.11}%
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\contentsline {subsection}{\numberline {50.2.14}\textcolor {Chapter }{ProjectiveSpecialUnitaryGroup}}{834}{subsection.50.2.14}%
\contentsline {subsection}{\numberline {50.2.15}\textcolor {Chapter }{ProjectiveSymplecticGroup}}{835}{subsection.50.2.15}%
\contentsline {subsection}{\numberline {50.2.16}\textcolor {Chapter }{ProjectiveGeneralOrthogonalGroup}}{835}{subsection.50.2.16}%
\contentsline {subsection}{\numberline {50.2.17}\textcolor {Chapter }{ProjectiveSpecialOrthogonalGroup}}{835}{subsection.50.2.17}%
\contentsline {subsection}{\numberline {50.2.18}\textcolor {Chapter }{ProjectiveOmega}}{835}{subsection.50.2.18}%
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\contentsline {section}{\numberline {50.6}\textcolor {Chapter }{Finite Perfect Groups}}{839}{section.50.6}%
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\contentsline {subsection}{\numberline {50.6.2}\textcolor {Chapter }{PerfectGroup}}{839}{subsection.50.6.2}%
\contentsline {subsection}{\numberline {50.6.3}\textcolor {Chapter }{PerfectIdentification}}{840}{subsection.50.6.3}%
\contentsline {subsection}{\numberline {50.6.4}\textcolor {Chapter }{NumberPerfectGroups}}{840}{subsection.50.6.4}%
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\contentsline {subsection}{\numberline {50.6.7}\textcolor {Chapter }{More about the Perfect Groups Library}}{842}{subsection.50.6.7}%
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\contentsline {chapter}{\numberline {51}\textcolor {Chapter }{Semigroups and Monoids}}{853}{chapter.51}%
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\contentsline {subsection}{\numberline {51.1.2}\textcolor {Chapter }{Semigroup}}{853}{subsection.51.1.2}%
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\contentsline {subsection}{\numberline {51.1.4}\textcolor {Chapter }{IsSubsemigroup}}{854}{subsection.51.1.4}%
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\contentsline {subsection}{\numberline {51.2.1}\textcolor {Chapter }{IsMonoid}}{857}{subsection.51.2.1}%
\contentsline {subsection}{\numberline {51.2.2}\textcolor {Chapter }{Monoid}}{858}{subsection.51.2.2}%
\contentsline {subsection}{\numberline {51.2.3}\textcolor {Chapter }{Submonoid}}{858}{subsection.51.2.3}%
\contentsline {subsection}{\numberline {51.2.4}\textcolor {Chapter }{MonoidByGenerators}}{858}{subsection.51.2.4}%
\contentsline {subsection}{\numberline {51.2.5}\textcolor {Chapter }{AsMonoid}}{858}{subsection.51.2.5}%
\contentsline {subsection}{\numberline {51.2.6}\textcolor {Chapter }{AsSubmonoid}}{858}{subsection.51.2.6}%
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\contentsline {subsection}{\numberline {51.2.9}\textcolor {Chapter }{FreeMonoid}}{859}{subsection.51.2.9}%
\contentsline {subsection}{\numberline {51.2.10}\textcolor {Chapter }{MonoidByMultiplicationTable}}{860}{subsection.51.2.10}%
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\contentsline {subsection}{\numberline {51.3.2}\textcolor {Chapter }{InverseMonoid}}{861}{subsection.51.3.2}%
\contentsline {subsection}{\numberline {51.3.3}\textcolor {Chapter }{GeneratorsOfInverseSemigroup}}{861}{subsection.51.3.3}%
\contentsline {subsection}{\numberline {51.3.4}\textcolor {Chapter }{GeneratorsOfInverseMonoid}}{862}{subsection.51.3.4}%
\contentsline {subsection}{\numberline {51.3.5}\textcolor {Chapter }{IsInverseSubsemigroup}}{862}{subsection.51.3.5}%
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\contentsline {subsection}{\numberline {51.4.2}\textcolor {Chapter }{IsRegularSemigroupElement}}{863}{subsection.51.4.2}%
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\contentsline {subsection}{\numberline {51.4.5}\textcolor {Chapter }{IsZeroSimpleSemigroup}}{864}{subsection.51.4.5}%
\contentsline {subsection}{\numberline {51.4.6}\textcolor {Chapter }{IsZeroGroup}}{864}{subsection.51.4.6}%
\contentsline {subsection}{\numberline {51.4.7}\textcolor {Chapter }{IsReesCongruenceSemigroup}}{864}{subsection.51.4.7}%
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\contentsline {subsection}{\numberline {51.5.2}\textcolor {Chapter }{ReesCongruenceOfSemigroupIdeal}}{865}{subsection.51.5.2}%
\contentsline {subsection}{\numberline {51.5.3}\textcolor {Chapter }{IsLeftSemigroupIdeal}}{865}{subsection.51.5.3}%
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\contentsline {subsection}{\numberline {51.6.2}\textcolor {Chapter }{IsReesCongruence}}{865}{subsection.51.6.2}%
\contentsline {section}{\numberline {51.7}\textcolor {Chapter }{Quotients}}{865}{section.51.7}%
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\contentsline {subsection}{\numberline {51.7.2}\textcolor {Chapter }{HomomorphismQuotientSemigroup}}{866}{subsection.51.7.2}%
\contentsline {subsection}{\numberline {51.7.3}\textcolor {Chapter }{QuotientSemigroupPreimage}}{866}{subsection.51.7.3}%
\contentsline {section}{\numberline {51.8}\textcolor {Chapter }{Green's Relations}}{866}{section.51.8}%
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\contentsline {subsection}{\numberline {51.8.2}\textcolor {Chapter }{IsGreensRelation}}{867}{subsection.51.8.2}%
\contentsline {subsection}{\numberline {51.8.3}\textcolor {Chapter }{IsGreensClass}}{867}{subsection.51.8.3}%
\contentsline {subsection}{\numberline {51.8.4}\textcolor {Chapter }{IsGreensLessThanOrEqual}}{867}{subsection.51.8.4}%
\contentsline {subsection}{\numberline {51.8.5}\textcolor {Chapter }{RClassOfHClass}}{867}{subsection.51.8.5}%
\contentsline {subsection}{\numberline {51.8.6}\textcolor {Chapter }{EggBoxOfDClass}}{867}{subsection.51.8.6}%
\contentsline {subsection}{\numberline {51.8.7}\textcolor {Chapter }{DisplayEggBoxOfDClass}}{868}{subsection.51.8.7}%
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\contentsline {subsection}{\numberline {51.8.9}\textcolor {Chapter }{GreensRClasses}}{868}{subsection.51.8.9}%
\contentsline {subsection}{\numberline {51.8.10}\textcolor {Chapter }{GroupHClassOfGreensDClass}}{868}{subsection.51.8.10}%
\contentsline {subsection}{\numberline {51.8.11}\textcolor {Chapter }{IsGroupHClass}}{868}{subsection.51.8.11}%
\contentsline {subsection}{\numberline {51.8.12}\textcolor {Chapter }{IsRegularDClass}}{869}{subsection.51.8.12}%
\contentsline {subsection}{\numberline {51.8.13}\textcolor {Chapter }{DisplaySemigroup}}{869}{subsection.51.8.13}%
\contentsline {section}{\numberline {51.9}\textcolor {Chapter }{Rees Matrix Semigroups}}{869}{section.51.9}%
\contentsline {subsection}{\numberline {51.9.1}\textcolor {Chapter }{ReesMatrixSemigroup}}{870}{subsection.51.9.1}%
\contentsline {subsection}{\numberline {51.9.2}\textcolor {Chapter }{ReesMatrixSubsemigroup}}{871}{subsection.51.9.2}%
\contentsline {subsection}{\numberline {51.9.3}\textcolor {Chapter }{IsomorphismReesMatrixSemigroup}}{871}{subsection.51.9.3}%
\contentsline {subsection}{\numberline {51.9.4}\textcolor {Chapter }{IsReesMatrixSemigroupElement}}{872}{subsection.51.9.4}%
\contentsline {subsection}{\numberline {51.9.5}\textcolor {Chapter }{ReesMatrixSemigroupElement}}{873}{subsection.51.9.5}%
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\contentsline {subsection}{\numberline {51.9.10}\textcolor {Chapter }{UnderlyingSemigroup (for a Rees matrix semigroup)}}{876}{subsection.51.9.10}%
\contentsline {subsection}{\numberline {51.9.11}\textcolor {Chapter }{AssociatedReesMatrixSemigroupOfDClass}}{876}{subsection.51.9.11}%
\contentsline {chapter}{\numberline {52}\textcolor {Chapter }{Finitely Presented Semigroups and Monoids}}{878}{chapter.52}%
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\contentsline {subsection}{\numberline {52.1.4}\textcolor {Chapter }{FpGrpMonSmgOfFpGrpMonSmgElement}}{881}{subsection.52.1.4}%
\contentsline {section}{\numberline {52.2}\textcolor {Chapter }{Creating Finitely Presented Semigroups and Monoids}}{881}{section.52.2}%
\contentsline {subsection}{\numberline {52.2.1}\textcolor {Chapter }{\texttt {\symbol {92}}/ (for a free semigroup or monoid and a list of pairs of elements)}}{881}{subsection.52.2.1}%
\contentsline {subsection}{\numberline {52.2.2}\textcolor {Chapter }{FactorFreeSemigroupByRelations}}{881}{subsection.52.2.2}%
\contentsline {subsection}{\numberline {52.2.3}\textcolor {Chapter }{IsomorphismFpSemigroup}}{882}{subsection.52.2.3}%
\contentsline {section}{\numberline {52.3}\textcolor {Chapter }{Comparison of Elements of Finitely Presented Semigroups}}{882}{section.52.3}%
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\contentsline {section}{\numberline {52.4}\textcolor {Chapter }{Preimages in the Free Semigroup or Monoid}}{882}{section.52.4}%
\contentsline {subsection}{\numberline {52.4.1}\textcolor {Chapter }{UnderlyingElement (of an element in a fp semigroup or monoid)}}{882}{subsection.52.4.1}%
\contentsline {subsection}{\numberline {52.4.2}\textcolor {Chapter }{ElementOfFpSemigroup}}{883}{subsection.52.4.2}%
\contentsline {subsection}{\numberline {52.4.3}\textcolor {Chapter }{FreeSemigroupOfFpSemigroup}}{883}{subsection.52.4.3}%
\contentsline {subsection}{\numberline {52.4.4}\textcolor {Chapter }{FreeGeneratorsOfFpSemigroup}}{883}{subsection.52.4.4}%
\contentsline {subsection}{\numberline {52.4.5}\textcolor {Chapter }{RelationsOfFpSemigroup}}{883}{subsection.52.4.5}%
\contentsline {section}{\numberline {52.5}\textcolor {Chapter }{Rewriting Systems and the Knuth\texttt {\symbol {45}}Bendix Procedure}}{884}{section.52.5}%
\contentsline {subsection}{\numberline {52.5.1}\textcolor {Chapter }{ReducedConfluentRewritingSystem}}{884}{subsection.52.5.1}%
\contentsline {subsection}{\numberline {52.5.2}\textcolor {Chapter }{KB{\textunderscore }REW}}{885}{subsection.52.5.2}%
\contentsline {subsection}{\numberline {52.5.3}\textcolor {Chapter }{KnuthBendixRewritingSystem}}{885}{subsection.52.5.3}%
\contentsline {subsection}{\numberline {52.5.4}\textcolor {Chapter }{SemigroupOfRewritingSystem}}{885}{subsection.52.5.4}%
\contentsline {subsection}{\numberline {52.5.5}\textcolor {Chapter }{FreeSemigroupOfRewritingSystem}}{886}{subsection.52.5.5}%
\contentsline {section}{\numberline {52.6}\textcolor {Chapter }{Todd\texttt {\symbol {45}}Coxeter Procedure}}{886}{section.52.6}%
\contentsline {subsection}{\numberline {52.6.1}\textcolor {Chapter }{CosetTableOfFpSemigroup}}{886}{subsection.52.6.1}%
\contentsline {chapter}{\numberline {53}\textcolor {Chapter }{Transformations}}{887}{chapter.53}%
\contentsline {section}{\numberline {53.1}\textcolor {Chapter }{The family and categories of transformations}}{888}{section.53.1}%
\contentsline {subsection}{\numberline {53.1.1}\textcolor {Chapter }{IsTransformation}}{888}{subsection.53.1.1}%
\contentsline {subsection}{\numberline {53.1.2}\textcolor {Chapter }{IsTransformationCollection}}{888}{subsection.53.1.2}%
\contentsline {subsection}{\numberline {53.1.3}\textcolor {Chapter }{TransformationFamily}}{889}{subsection.53.1.3}%
\contentsline {section}{\numberline {53.2}\textcolor {Chapter }{Creating transformations}}{889}{section.53.2}%
\contentsline {subsection}{\numberline {53.2.1}\textcolor {Chapter }{Transformation (for an image list)}}{889}{subsection.53.2.1}%
\contentsline {subsection}{\numberline {53.2.2}\textcolor {Chapter }{Transformation (for a source and destination)}}{889}{subsection.53.2.2}%
\contentsline {subsection}{\numberline {53.2.3}\textcolor {Chapter }{TransformationByImageAndKernel (for an image and kernel)}}{890}{subsection.53.2.3}%
\contentsline {subsection}{\numberline {53.2.4}\textcolor {Chapter }{Idempotent}}{890}{subsection.53.2.4}%
\contentsline {subsection}{\numberline {53.2.5}\textcolor {Chapter }{TransformationOp}}{890}{subsection.53.2.5}%
\contentsline {subsection}{\numberline {53.2.6}\textcolor {Chapter }{TransformationNumber}}{891}{subsection.53.2.6}%
\contentsline {subsection}{\numberline {53.2.7}\textcolor {Chapter }{RandomTransformation}}{892}{subsection.53.2.7}%
\contentsline {subsection}{\numberline {53.2.8}\textcolor {Chapter }{IdentityTransformation}}{892}{subsection.53.2.8}%
\contentsline {subsection}{\numberline {53.2.9}\textcolor {Chapter }{ConstantTransformation}}{892}{subsection.53.2.9}%
\contentsline {section}{\numberline {53.3}\textcolor {Chapter }{Changing the representation of a transformation}}{892}{section.53.3}%
\contentsline {subsection}{\numberline {53.3.1}\textcolor {Chapter }{AsTransformation}}{892}{subsection.53.3.1}%
\contentsline {subsection}{\numberline {53.3.2}\textcolor {Chapter }{RestrictedTransformation}}{894}{subsection.53.3.2}%
\contentsline {subsection}{\numberline {53.3.3}\textcolor {Chapter }{PermutationOfImage}}{894}{subsection.53.3.3}%
\contentsline {section}{\numberline {53.4}\textcolor {Chapter }{Operators for transformations}}{894}{section.53.4}%
\contentsline {subsection}{\numberline {53.4.1}\textcolor {Chapter }{\texttt {\symbol {92}}\texttt {\symbol {94}} (for a positive integer and a transformation)}}{894}{subsection.53.4.1}%
\contentsline {subsection}{\numberline {53.4.2}\textcolor {Chapter }{\texttt {\symbol {92}}\texttt {\symbol {94}} (for a transformation and a permutation)}}{894}{subsection.53.4.2}%
\contentsline {subsection}{\numberline {53.4.3}\textcolor {Chapter }{\texttt {\symbol {92}}* (for transformations)}}{895}{subsection.53.4.3}%
\contentsline {subsection}{\numberline {53.4.4}\textcolor {Chapter }{\texttt {\symbol {92}}/ (for a transformation and a permutation)}}{895}{subsection.53.4.4}%
\contentsline {subsection}{\numberline {53.4.5}\textcolor {Chapter }{LeftQuotient (for a permutation and transformation)}}{895}{subsection.53.4.5}%
\contentsline {subsection}{\numberline {53.4.6}\textcolor {Chapter }{\texttt {\symbol {92}}{\textless } (for transformations)}}{895}{subsection.53.4.6}%
\contentsline {subsection}{\numberline {53.4.7}\textcolor {Chapter }{\texttt {\symbol {92}}= (for transformations)}}{895}{subsection.53.4.7}%
\contentsline {subsection}{\numberline {53.4.8}\textcolor {Chapter }{PermLeftQuoTransformation}}{895}{subsection.53.4.8}%
\contentsline {subsection}{\numberline {53.4.9}\textcolor {Chapter }{IsInjectiveListTrans}}{896}{subsection.53.4.9}%
\contentsline {subsection}{\numberline {53.4.10}\textcolor {Chapter }{ComponentTransformationInt}}{896}{subsection.53.4.10}%
\contentsline {subsection}{\numberline {53.4.11}\textcolor {Chapter }{PreImagesOfTransformation}}{896}{subsection.53.4.11}%
\contentsline {section}{\numberline {53.5}\textcolor {Chapter }{Attributes for transformations}}{897}{section.53.5}%
\contentsline {subsection}{\numberline {53.5.1}\textcolor {Chapter }{DegreeOfTransformation}}{897}{subsection.53.5.1}%
\contentsline {subsection}{\numberline {53.5.2}\textcolor {Chapter }{ImageListOfTransformation}}{897}{subsection.53.5.2}%
\contentsline {subsection}{\numberline {53.5.3}\textcolor {Chapter }{ImageSetOfTransformation}}{898}{subsection.53.5.3}%
\contentsline {subsection}{\numberline {53.5.4}\textcolor {Chapter }{RankOfTransformation (for a transformation and a positive integer)}}{898}{subsection.53.5.4}%
\contentsline {subsection}{\numberline {53.5.5}\textcolor {Chapter }{MovedPoints (for a transformation)}}{898}{subsection.53.5.5}%
\contentsline {subsection}{\numberline {53.5.6}\textcolor {Chapter }{NrMovedPoints (for a transformation)}}{899}{subsection.53.5.6}%
\contentsline {subsection}{\numberline {53.5.7}\textcolor {Chapter }{SmallestMovedPoint (for a transformation)}}{899}{subsection.53.5.7}%
\contentsline {subsection}{\numberline {53.5.8}\textcolor {Chapter }{LargestMovedPoint (for a transformation)}}{899}{subsection.53.5.8}%
\contentsline {subsection}{\numberline {53.5.9}\textcolor {Chapter }{SmallestImageOfMovedPoint (for a transformation)}}{900}{subsection.53.5.9}%
\contentsline {subsection}{\numberline {53.5.10}\textcolor {Chapter }{LargestImageOfMovedPoint (for a transformation)}}{900}{subsection.53.5.10}%
\contentsline {subsection}{\numberline {53.5.11}\textcolor {Chapter }{FlatKernelOfTransformation}}{901}{subsection.53.5.11}%
\contentsline {subsection}{\numberline {53.5.12}\textcolor {Chapter }{KernelOfTransformation}}{901}{subsection.53.5.12}%
\contentsline {subsection}{\numberline {53.5.13}\textcolor {Chapter }{InverseOfTransformation}}{902}{subsection.53.5.13}%
\contentsline {subsection}{\numberline {53.5.14}\textcolor {Chapter }{Inverse (for a transformation)}}{902}{subsection.53.5.14}%
\contentsline {subsection}{\numberline {53.5.15}\textcolor {Chapter }{IndexPeriodOfTransformation}}{902}{subsection.53.5.15}%
\contentsline {subsection}{\numberline {53.5.16}\textcolor {Chapter }{SmallestIdempotentPower (for a transformation)}}{903}{subsection.53.5.16}%
\contentsline {subsection}{\numberline {53.5.17}\textcolor {Chapter }{ComponentsOfTransformation}}{903}{subsection.53.5.17}%
\contentsline {subsection}{\numberline {53.5.18}\textcolor {Chapter }{NrComponentsOfTransformation}}{903}{subsection.53.5.18}%
\contentsline {subsection}{\numberline {53.5.19}\textcolor {Chapter }{ComponentRepsOfTransformation}}{904}{subsection.53.5.19}%
\contentsline {subsection}{\numberline {53.5.20}\textcolor {Chapter }{CyclesOfTransformation}}{904}{subsection.53.5.20}%
\contentsline {subsection}{\numberline {53.5.21}\textcolor {Chapter }{CycleTransformationInt}}{904}{subsection.53.5.21}%
\contentsline {subsection}{\numberline {53.5.22}\textcolor {Chapter }{LeftOne (for a transformation)}}{905}{subsection.53.5.22}%
\contentsline {subsection}{\numberline {53.5.23}\textcolor {Chapter }{TrimTransformation}}{905}{subsection.53.5.23}%
\contentsline {section}{\numberline {53.6}\textcolor {Chapter }{Displaying transformations}}{906}{section.53.6}%
\contentsline {section}{\numberline {53.7}\textcolor {Chapter }{Semigroups of transformations}}{907}{section.53.7}%
\contentsline {subsection}{\numberline {53.7.1}\textcolor {Chapter }{IsTransformationSemigroup}}{907}{subsection.53.7.1}%
\contentsline {subsection}{\numberline {53.7.2}\textcolor {Chapter }{DegreeOfTransformationSemigroup}}{908}{subsection.53.7.2}%
\contentsline {subsection}{\numberline {53.7.3}\textcolor {Chapter }{FullTransformationSemigroup}}{908}{subsection.53.7.3}%
\contentsline {subsection}{\numberline {53.7.4}\textcolor {Chapter }{IsFullTransformationSemigroup}}{908}{subsection.53.7.4}%
\contentsline {subsection}{\numberline {53.7.5}\textcolor {Chapter }{IsomorphismTransformationSemigroup}}{909}{subsection.53.7.5}%
\contentsline {subsection}{\numberline {53.7.6}\textcolor {Chapter }{AntiIsomorphismTransformationSemigroup}}{910}{subsection.53.7.6}%
\contentsline {chapter}{\numberline {54}\textcolor {Chapter }{Partial permutations}}{911}{chapter.54}%
\contentsline {section}{\numberline {54.1}\textcolor {Chapter }{The family and categories of partial permutations}}{913}{section.54.1}%
\contentsline {subsection}{\numberline {54.1.1}\textcolor {Chapter }{IsPartialPerm}}{913}{subsection.54.1.1}%
\contentsline {subsection}{\numberline {54.1.2}\textcolor {Chapter }{IsPartialPermCollection}}{913}{subsection.54.1.2}%
\contentsline {subsection}{\numberline {54.1.3}\textcolor {Chapter }{PartialPermFamily}}{913}{subsection.54.1.3}%
\contentsline {section}{\numberline {54.2}\textcolor {Chapter }{Creating partial permutations}}{913}{section.54.2}%
\contentsline {subsection}{\numberline {54.2.1}\textcolor {Chapter }{PartialPerm (for a domain and image)}}{913}{subsection.54.2.1}%
\contentsline {subsection}{\numberline {54.2.2}\textcolor {Chapter }{PartialPermOp}}{914}{subsection.54.2.2}%
\contentsline {subsection}{\numberline {54.2.3}\textcolor {Chapter }{RestrictedPartialPerm}}{915}{subsection.54.2.3}%
\contentsline {subsection}{\numberline {54.2.4}\textcolor {Chapter }{JoinOfPartialPerms}}{915}{subsection.54.2.4}%
\contentsline {subsection}{\numberline {54.2.5}\textcolor {Chapter }{MeetOfPartialPerms}}{916}{subsection.54.2.5}%
\contentsline {subsection}{\numberline {54.2.6}\textcolor {Chapter }{EmptyPartialPerm}}{916}{subsection.54.2.6}%
\contentsline {subsection}{\numberline {54.2.7}\textcolor {Chapter }{RandomPartialPerm}}{917}{subsection.54.2.7}%
\contentsline {section}{\numberline {54.3}\textcolor {Chapter }{Attributes for partial permutations}}{917}{section.54.3}%
\contentsline {subsection}{\numberline {54.3.1}\textcolor {Chapter }{DegreeOfPartialPerm}}{917}{subsection.54.3.1}%
\contentsline {subsection}{\numberline {54.3.2}\textcolor {Chapter }{CodegreeOfPartialPerm}}{918}{subsection.54.3.2}%
\contentsline {subsection}{\numberline {54.3.3}\textcolor {Chapter }{RankOfPartialPerm}}{918}{subsection.54.3.3}%
\contentsline {subsection}{\numberline {54.3.4}\textcolor {Chapter }{DomainOfPartialPerm}}{918}{subsection.54.3.4}%
\contentsline {subsection}{\numberline {54.3.5}\textcolor {Chapter }{ImageOfPartialPermCollection}}{918}{subsection.54.3.5}%
\contentsline {subsection}{\numberline {54.3.6}\textcolor {Chapter }{ImageListOfPartialPerm}}{919}{subsection.54.3.6}%
\contentsline {subsection}{\numberline {54.3.7}\textcolor {Chapter }{ImageSetOfPartialPerm}}{919}{subsection.54.3.7}%
\contentsline {subsection}{\numberline {54.3.8}\textcolor {Chapter }{FixedPointsOfPartialPerm (for a partial perm)}}{919}{subsection.54.3.8}%
\contentsline {subsection}{\numberline {54.3.9}\textcolor {Chapter }{MovedPoints (for a partial perm)}}{920}{subsection.54.3.9}%
\contentsline {subsection}{\numberline {54.3.10}\textcolor {Chapter }{NrFixedPoints (for a partial perm)}}{920}{subsection.54.3.10}%
\contentsline {subsection}{\numberline {54.3.11}\textcolor {Chapter }{NrMovedPoints (for a partial perm)}}{920}{subsection.54.3.11}%
\contentsline {subsection}{\numberline {54.3.12}\textcolor {Chapter }{SmallestMovedPoint (for a partial perm)}}{921}{subsection.54.3.12}%
\contentsline {subsection}{\numberline {54.3.13}\textcolor {Chapter }{LargestMovedPoint (for a partial perm)}}{921}{subsection.54.3.13}%
\contentsline {subsection}{\numberline {54.3.14}\textcolor {Chapter }{SmallestImageOfMovedPoint (for a partial permutation)}}{921}{subsection.54.3.14}%
\contentsline {subsection}{\numberline {54.3.15}\textcolor {Chapter }{LargestImageOfMovedPoint (for a partial permutation)}}{922}{subsection.54.3.15}%
\contentsline {subsection}{\numberline {54.3.16}\textcolor {Chapter }{IndexPeriodOfPartialPerm}}{922}{subsection.54.3.16}%
\contentsline {subsection}{\numberline {54.3.17}\textcolor {Chapter }{SmallestIdempotentPower (for a partial perm)}}{923}{subsection.54.3.17}%
\contentsline {subsection}{\numberline {54.3.18}\textcolor {Chapter }{ComponentsOfPartialPerm}}{923}{subsection.54.3.18}%
\contentsline {subsection}{\numberline {54.3.19}\textcolor {Chapter }{NrComponentsOfPartialPerm}}{923}{subsection.54.3.19}%
\contentsline {subsection}{\numberline {54.3.20}\textcolor {Chapter }{ComponentRepsOfPartialPerm}}{924}{subsection.54.3.20}%
\contentsline {subsection}{\numberline {54.3.21}\textcolor {Chapter }{LeftOne (for a partial perm)}}{924}{subsection.54.3.21}%
\contentsline {subsection}{\numberline {54.3.22}\textcolor {Chapter }{One (for a partial perm)}}{924}{subsection.54.3.22}%
\contentsline {subsection}{\numberline {54.3.23}\textcolor {Chapter }{MultiplicativeZero (for a partial perm)}}{925}{subsection.54.3.23}%
\contentsline {section}{\numberline {54.4}\textcolor {Chapter }{Changing the representation of a partial permutation}}{925}{section.54.4}%
\contentsline {subsection}{\numberline {54.4.1}\textcolor {Chapter }{AsPartialPerm (for a permutation and a set of positive integers)}}{925}{subsection.54.4.1}%
\contentsline {subsection}{\numberline {54.4.2}\textcolor {Chapter }{AsPartialPerm (for a transformation and a set of positive integer)}}{926}{subsection.54.4.2}%
\contentsline {section}{\numberline {54.5}\textcolor {Chapter }{Operators and operations for partial permutations}}{926}{section.54.5}%
\contentsline {subsection}{\numberline {54.5.1}\textcolor {Chapter }{Inverse (for a partial permutation)}}{926}{subsection.54.5.1}%
\contentsline {subsection}{\numberline {54.5.2}\textcolor {Chapter }{\texttt {\symbol {92}}\texttt {\symbol {94}} (for a positive integer and a partial permutation)}}{927}{subsection.54.5.2}%
\contentsline {subsection}{\numberline {54.5.3}\textcolor {Chapter }{\texttt {\symbol {92}}/ (for a positive integer and a partial permutation)}}{927}{subsection.54.5.3}%
\contentsline {subsection}{\numberline {54.5.4}\textcolor {Chapter }{\texttt {\symbol {92}}\texttt {\symbol {94}} (for a partial permutation and a permutation or partial permutation)}}{927}{subsection.54.5.4}%
\contentsline {subsection}{\numberline {54.5.5}\textcolor {Chapter }{\texttt {\symbol {92}}* (for permutations and partial permutations)}}{927}{subsection.54.5.5}%
\contentsline {subsection}{\numberline {54.5.6}\textcolor {Chapter }{\texttt {\symbol {92}}/ (for a partial permutation and permutation or partial permutation)}}{927}{subsection.54.5.6}%
\contentsline {subsection}{\numberline {54.5.7}\textcolor {Chapter }{LeftQuotient (for a permutation or partial permutation and a partial permutation)}}{927}{subsection.54.5.7}%
\contentsline {subsection}{\numberline {54.5.8}\textcolor {Chapter }{\texttt {\symbol {92}}{\textless } (for partial permutations)}}{928}{subsection.54.5.8}%
\contentsline {subsection}{\numberline {54.5.9}\textcolor {Chapter }{\texttt {\symbol {92}}= (for partial permutations)}}{928}{subsection.54.5.9}%
\contentsline {subsection}{\numberline {54.5.10}\textcolor {Chapter }{PermLeftQuoPartialPerm}}{928}{subsection.54.5.10}%
\contentsline {subsection}{\numberline {54.5.11}\textcolor {Chapter }{PreImagePartialPerm}}{928}{subsection.54.5.11}%
\contentsline {subsection}{\numberline {54.5.12}\textcolor {Chapter }{ComponentPartialPermInt}}{929}{subsection.54.5.12}%
\contentsline {subsection}{\numberline {54.5.13}\textcolor {Chapter }{NaturalLeqPartialPerm}}{929}{subsection.54.5.13}%
\contentsline {subsection}{\numberline {54.5.14}\textcolor {Chapter }{ShortLexLeqPartialPerm}}{930}{subsection.54.5.14}%
\contentsline {subsection}{\numberline {54.5.15}\textcolor {Chapter }{TrimPartialPerm}}{930}{subsection.54.5.15}%
\contentsline {section}{\numberline {54.6}\textcolor {Chapter }{Displaying partial permutations}}{931}{section.54.6}%
\contentsline {section}{\numberline {54.7}\textcolor {Chapter }{Semigroups and inverse semigroups of partial permutations}}{932}{section.54.7}%
\contentsline {subsection}{\numberline {54.7.1}\textcolor {Chapter }{IsPartialPermSemigroup}}{933}{subsection.54.7.1}%
\contentsline {subsection}{\numberline {54.7.2}\textcolor {Chapter }{DegreeOfPartialPermSemigroup}}{933}{subsection.54.7.2}%
\contentsline {subsection}{\numberline {54.7.3}\textcolor {Chapter }{SymmetricInverseSemigroup}}{933}{subsection.54.7.3}%
\contentsline {subsection}{\numberline {54.7.4}\textcolor {Chapter }{IsSymmetricInverseSemigroup}}{934}{subsection.54.7.4}%
\contentsline {subsection}{\numberline {54.7.5}\textcolor {Chapter }{NaturalPartialOrder}}{934}{subsection.54.7.5}%
\contentsline {subsection}{\numberline {54.7.6}\textcolor {Chapter }{IsomorphismPartialPermSemigroup}}{935}{subsection.54.7.6}%
\contentsline {chapter}{\numberline {55}\textcolor {Chapter }{Additive Magmas}}{937}{chapter.55}%
\contentsline {section}{\numberline {55.1}\textcolor {Chapter }{(Near\texttt {\symbol {45}})Additive Magma Categories}}{937}{section.55.1}%
\contentsline {subsection}{\numberline {55.1.1}\textcolor {Chapter }{IsNearAdditiveMagma}}{937}{subsection.55.1.1}%
\contentsline {subsection}{\numberline {55.1.2}\textcolor {Chapter }{IsNearAdditiveMagmaWithZero}}{937}{subsection.55.1.2}%
\contentsline {subsection}{\numberline {55.1.3}\textcolor {Chapter }{IsNearAdditiveGroup}}{937}{subsection.55.1.3}%
\contentsline {subsection}{\numberline {55.1.4}\textcolor {Chapter }{IsAdditiveMagma}}{938}{subsection.55.1.4}%
\contentsline {subsection}{\numberline {55.1.5}\textcolor {Chapter }{IsAdditiveMagmaWithZero}}{938}{subsection.55.1.5}%
\contentsline {subsection}{\numberline {55.1.6}\textcolor {Chapter }{IsAdditiveGroup}}{938}{subsection.55.1.6}%
\contentsline {section}{\numberline {55.2}\textcolor {Chapter }{(Near\texttt {\symbol {45}})Additive Magma Generation}}{939}{section.55.2}%
\contentsline {subsection}{\numberline {55.2.1}\textcolor {Chapter }{NearAdditiveMagma}}{939}{subsection.55.2.1}%
\contentsline {subsection}{\numberline {55.2.2}\textcolor {Chapter }{NearAdditiveMagmaWithZero}}{939}{subsection.55.2.2}%
\contentsline {subsection}{\numberline {55.2.3}\textcolor {Chapter }{NearAdditiveGroup}}{939}{subsection.55.2.3}%
\contentsline {subsection}{\numberline {55.2.4}\textcolor {Chapter }{NearAdditiveMagmaByGenerators}}{939}{subsection.55.2.4}%
\contentsline {subsection}{\numberline {55.2.5}\textcolor {Chapter }{NearAdditiveMagmaWithZeroByGenerators}}{939}{subsection.55.2.5}%
\contentsline {subsection}{\numberline {55.2.6}\textcolor {Chapter }{NearAdditiveGroupByGenerators}}{940}{subsection.55.2.6}%
\contentsline {subsection}{\numberline {55.2.7}\textcolor {Chapter }{SubnearAdditiveMagma}}{940}{subsection.55.2.7}%
\contentsline {subsection}{\numberline {55.2.8}\textcolor {Chapter }{SubnearAdditiveMagmaWithZero}}{940}{subsection.55.2.8}%
\contentsline {subsection}{\numberline {55.2.9}\textcolor {Chapter }{SubnearAdditiveGroup}}{940}{subsection.55.2.9}%
\contentsline {section}{\numberline {55.3}\textcolor {Chapter }{Attributes and Properties for (Near\texttt {\symbol {45}})Additive Magmas}}{941}{section.55.3}%
\contentsline {subsection}{\numberline {55.3.1}\textcolor {Chapter }{IsAdditivelyCommutative}}{941}{subsection.55.3.1}%
\contentsline {subsection}{\numberline {55.3.2}\textcolor {Chapter }{GeneratorsOfNearAdditiveMagma}}{941}{subsection.55.3.2}%
\contentsline {subsection}{\numberline {55.3.3}\textcolor {Chapter }{GeneratorsOfNearAdditiveMagmaWithZero}}{941}{subsection.55.3.3}%
\contentsline {subsection}{\numberline {55.3.4}\textcolor {Chapter }{GeneratorsOfNearAdditiveGroup}}{941}{subsection.55.3.4}%
\contentsline {subsection}{\numberline {55.3.5}\textcolor {Chapter }{AdditiveNeutralElement}}{941}{subsection.55.3.5}%
\contentsline {subsection}{\numberline {55.3.6}\textcolor {Chapter }{TrivialSubnearAdditiveMagmaWithZero}}{942}{subsection.55.3.6}%
\contentsline {section}{\numberline {55.4}\textcolor {Chapter }{Operations for (Near\texttt {\symbol {45}})Additive Magmas}}{942}{section.55.4}%
\contentsline {subsection}{\numberline {55.4.1}\textcolor {Chapter }{ClosureNearAdditiveGroup}}{942}{subsection.55.4.1}%
\contentsline {subsection}{\numberline {55.4.2}\textcolor {Chapter }{ShowAdditionTable}}{942}{subsection.55.4.2}%
\contentsline {chapter}{\numberline {56}\textcolor {Chapter }{Rings}}{943}{chapter.56}%
\contentsline {section}{\numberline {56.1}\textcolor {Chapter }{Generating Rings}}{943}{section.56.1}%
\contentsline {subsection}{\numberline {56.1.1}\textcolor {Chapter }{IsRing}}{943}{subsection.56.1.1}%
\contentsline {subsection}{\numberline {56.1.2}\textcolor {Chapter }{Ring}}{943}{subsection.56.1.2}%
\contentsline {subsection}{\numberline {56.1.3}\textcolor {Chapter }{DefaultRing}}{944}{subsection.56.1.3}%
\contentsline {subsection}{\numberline {56.1.4}\textcolor {Chapter }{RingByGenerators}}{944}{subsection.56.1.4}%
\contentsline {subsection}{\numberline {56.1.5}\textcolor {Chapter }{DefaultRingByGenerators}}{944}{subsection.56.1.5}%
\contentsline {subsection}{\numberline {56.1.6}\textcolor {Chapter }{GeneratorsOfRing}}{945}{subsection.56.1.6}%
\contentsline {subsection}{\numberline {56.1.7}\textcolor {Chapter }{Subring}}{945}{subsection.56.1.7}%
\contentsline {subsection}{\numberline {56.1.8}\textcolor {Chapter }{ClosureRing}}{945}{subsection.56.1.8}%
\contentsline {subsection}{\numberline {56.1.9}\textcolor {Chapter }{Quotient}}{945}{subsection.56.1.9}%
\contentsline {section}{\numberline {56.2}\textcolor {Chapter }{Ideals of Rings}}{946}{section.56.2}%
\contentsline {subsection}{\numberline {56.2.1}\textcolor {Chapter }{TwoSidedIdeal}}{946}{subsection.56.2.1}%
\contentsline {subsection}{\numberline {56.2.2}\textcolor {Chapter }{TwoSidedIdealNC}}{947}{subsection.56.2.2}%
\contentsline {subsection}{\numberline {56.2.3}\textcolor {Chapter }{IsTwoSidedIdeal}}{947}{subsection.56.2.3}%
\contentsline {subsection}{\numberline {56.2.4}\textcolor {Chapter }{TwoSidedIdealByGenerators}}{948}{subsection.56.2.4}%
\contentsline {subsection}{\numberline {56.2.5}\textcolor {Chapter }{LeftIdealByGenerators}}{948}{subsection.56.2.5}%
\contentsline {subsection}{\numberline {56.2.6}\textcolor {Chapter }{RightIdealByGenerators}}{948}{subsection.56.2.6}%
\contentsline {subsection}{\numberline {56.2.7}\textcolor {Chapter }{GeneratorsOfTwoSidedIdeal}}{948}{subsection.56.2.7}%
\contentsline {subsection}{\numberline {56.2.8}\textcolor {Chapter }{GeneratorsOfLeftIdeal}}{948}{subsection.56.2.8}%
\contentsline {subsection}{\numberline {56.2.9}\textcolor {Chapter }{GeneratorsOfRightIdeal}}{949}{subsection.56.2.9}%
\contentsline {subsection}{\numberline {56.2.10}\textcolor {Chapter }{LeftActingRingOfIdeal}}{949}{subsection.56.2.10}%
\contentsline {subsection}{\numberline {56.2.11}\textcolor {Chapter }{AsLeftIdeal}}{949}{subsection.56.2.11}%
\contentsline {section}{\numberline {56.3}\textcolor {Chapter }{Rings With One}}{949}{section.56.3}%
\contentsline {subsection}{\numberline {56.3.1}\textcolor {Chapter }{IsRingWithOne}}{949}{subsection.56.3.1}%
\contentsline {subsection}{\numberline {56.3.2}\textcolor {Chapter }{RingWithOne}}{950}{subsection.56.3.2}%
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\contentsline {subsection}{\numberline {64.12.2}\textcolor {Chapter }{Cochain}}{1077}{subsection.64.12.2}%
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\contentsline {subsection}{\numberline {64.12.7}\textcolor {Chapter }{Coboundaries}}{1078}{subsection.64.12.7}%
\contentsline {section}{\numberline {64.13}\textcolor {Chapter }{Modules over Semisimple Lie Algebras}}{1079}{section.64.13}%
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\contentsline {chapter}{\numberline {65}\textcolor {Chapter }{Magma Rings}}{1085}{chapter.65}%
\contentsline {section}{\numberline {65.1}\textcolor {Chapter }{Free Magma Rings}}{1086}{section.65.1}%
\contentsline {subsection}{\numberline {65.1.1}\textcolor {Chapter }{FreeMagmaRing}}{1086}{subsection.65.1.1}%
\contentsline {subsection}{\numberline {65.1.2}\textcolor {Chapter }{GroupRing}}{1086}{subsection.65.1.2}%
\contentsline {subsection}{\numberline {65.1.3}\textcolor {Chapter }{IsFreeMagmaRing}}{1086}{subsection.65.1.3}%
\contentsline {subsection}{\numberline {65.1.4}\textcolor {Chapter }{IsFreeMagmaRingWithOne}}{1087}{subsection.65.1.4}%
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\contentsline {subsection}{\numberline {65.1.6}\textcolor {Chapter }{UnderlyingMagma}}{1087}{subsection.65.1.6}%
\contentsline {subsection}{\numberline {65.1.7}\textcolor {Chapter }{AugmentationIdeal}}{1087}{subsection.65.1.7}%
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\contentsline {subsection}{\numberline {65.2.6}\textcolor {Chapter }{ElementOfMagmaRing}}{1088}{subsection.65.2.6}%
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\contentsline {section}{\numberline {65.4}\textcolor {Chapter }{Magma Rings modulo Relations}}{1089}{section.65.4}%
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\contentsline {chapter}{\numberline {66}\textcolor {Chapter }{Polynomials and Rational Functions}}{1092}{chapter.66}%
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\contentsline {subsection}{\numberline {66.4.2}\textcolor {Chapter }{NumeratorOfRationalFunction}}{1097}{subsection.66.4.2}%
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\contentsline {subsection}{\numberline {66.4.9}\textcolor {Chapter }{CoefficientsOfUnivariatePolynomial}}{1099}{subsection.66.4.9}%
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\contentsline {subsection}{\numberline {66.4.12}\textcolor {Chapter }{IsPrimitivePolynomial}}{1099}{subsection.66.4.12}%
\contentsline {subsection}{\numberline {66.4.13}\textcolor {Chapter }{SplittingField}}{1099}{subsection.66.4.13}%
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\contentsline {subsection}{\numberline {66.5.2}\textcolor {Chapter }{UnivariatePolynomialByCoefficients}}{1100}{subsection.66.5.2}%
\contentsline {subsection}{\numberline {66.5.3}\textcolor {Chapter }{DegreeOfLaurentPolynomial}}{1100}{subsection.66.5.3}%
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\contentsline {subsection}{\numberline {66.5.5}\textcolor {Chapter }{RootsOfUPol}}{1101}{subsection.66.5.5}%
\contentsline {subsection}{\numberline {66.5.6}\textcolor {Chapter }{QuotRemLaurpols}}{1101}{subsection.66.5.6}%
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\contentsline {subsection}{\numberline {66.5.8}\textcolor {Chapter }{InfoPoly}}{1102}{subsection.66.5.8}%
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\contentsline {subsection}{\numberline {66.6.3}\textcolor {Chapter }{LeadingCoefficient}}{1103}{subsection.66.6.3}%
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\contentsline {subsection}{\numberline {66.6.6}\textcolor {Chapter }{Discriminant}}{1103}{subsection.66.6.6}%
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\contentsline {section}{\numberline {66.8}\textcolor {Chapter }{Minimal Polynomials}}{1105}{section.66.8}%
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\contentsline {section}{\numberline {66.9}\textcolor {Chapter }{Cyclotomic Polynomials}}{1105}{section.66.9}%
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\contentsline {section}{\numberline {66.10}\textcolor {Chapter }{Polynomial Factorization}}{1105}{section.66.10}%
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\contentsline {subsection}{\numberline {66.10.2}\textcolor {Chapter }{FactorsSquarefree}}{1106}{subsection.66.10.2}%
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\contentsline {subsection}{\numberline {66.11.2}\textcolor {Chapter }{PolynomialModP}}{1106}{subsection.66.11.2}%
\contentsline {subsection}{\numberline {66.11.3}\textcolor {Chapter }{GaloisType}}{1107}{subsection.66.11.3}%
\contentsline {subsection}{\numberline {66.11.4}\textcolor {Chapter }{ProbabilityShapes}}{1107}{subsection.66.11.4}%
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\contentsline {subsection}{\numberline {66.12.4}\textcolor {Chapter }{OneFactorBound}}{1108}{subsection.66.12.4}%
\contentsline {section}{\numberline {66.13}\textcolor {Chapter }{Laurent Polynomials}}{1108}{section.66.13}%
\contentsline {subsection}{\numberline {66.13.1}\textcolor {Chapter }{LaurentPolynomialByCoefficients}}{1108}{subsection.66.13.1}%
\contentsline {subsection}{\numberline {66.13.2}\textcolor {Chapter }{CoefficientsOfLaurentPolynomial}}{1108}{subsection.66.13.2}%
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\contentsline {section}{\numberline {66.14}\textcolor {Chapter }{Univariate Rational Functions}}{1109}{section.66.14}%
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\contentsline {subsection}{\numberline {66.15.9}\textcolor {Chapter }{IsFunctionField}}{1112}{subsection.66.15.9}%
\contentsline {section}{\numberline {66.16}\textcolor {Chapter }{Univariate Polynomial Rings}}{1112}{section.66.16}%
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\contentsline {subsection}{\numberline {66.16.2}\textcolor {Chapter }{IsUnivariatePolynomialRing}}{1113}{subsection.66.16.2}%
\contentsline {section}{\numberline {66.17}\textcolor {Chapter }{Monomial Orderings}}{1113}{section.66.17}%
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\contentsline {subsection}{\numberline {66.17.2}\textcolor {Chapter }{LeadingMonomialOfPolynomial}}{1113}{subsection.66.17.2}%
\contentsline {subsection}{\numberline {66.17.3}\textcolor {Chapter }{LeadingTermOfPolynomial}}{1114}{subsection.66.17.3}%
\contentsline {subsection}{\numberline {66.17.4}\textcolor {Chapter }{LeadingCoefficientOfPolynomial}}{1114}{subsection.66.17.4}%
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\contentsline {subsection}{\numberline {66.17.6}\textcolor {Chapter }{MonomialExtrepComparisonFun}}{1115}{subsection.66.17.6}%
\contentsline {subsection}{\numberline {66.17.7}\textcolor {Chapter }{MonomialLexOrdering}}{1115}{subsection.66.17.7}%
\contentsline {subsection}{\numberline {66.17.8}\textcolor {Chapter }{MonomialGrlexOrdering}}{1115}{subsection.66.17.8}%
\contentsline {subsection}{\numberline {66.17.9}\textcolor {Chapter }{MonomialGrevlexOrdering}}{1116}{subsection.66.17.9}%
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\contentsline {subsection}{\numberline {66.17.11}\textcolor {Chapter }{PolynomialReduction}}{1116}{subsection.66.17.11}%
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\contentsline {subsection}{\numberline {66.17.14}\textcolor {Chapter }{MonomialExtGrlexLess}}{1117}{subsection.66.17.14}%
\contentsline {section}{\numberline {66.18}\textcolor {Chapter }{Groebner Bases}}{1117}{section.66.18}%
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\contentsline {subsection}{\numberline {66.18.2}\textcolor {Chapter }{ReducedGroebnerBasis}}{1118}{subsection.66.18.2}%
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\contentsline {subsection}{\numberline {66.23.2}\textcolor {Chapter }{ZippedProduct}}{1123}{subsection.66.23.2}%
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\contentsline {chapter}{\numberline {67}\textcolor {Chapter }{Algebraic extensions of fields}}{1125}{chapter.67}%
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\contentsline {subsection}{\numberline {67.1.2}\textcolor {Chapter }{IsAlgebraicExtension}}{1126}{subsection.67.1.2}%
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\contentsline {chapter}{\numberline {68}\textcolor {Chapter }{p\texttt {\symbol {45}}adic Numbers (preliminary)}}{1128}{chapter.68}%
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\contentsline {chapter}{\numberline {69}\textcolor {Chapter }{The MeatAxe}}{1132}{chapter.69}%
\contentsline {section}{\numberline {69.1}\textcolor {Chapter }{MeatAxe Modules}}{1132}{section.69.1}%
\contentsline {subsection}{\numberline {69.1.1}\textcolor {Chapter }{GModuleByMats}}{1132}{subsection.69.1.1}%
\contentsline {section}{\numberline {69.2}\textcolor {Chapter }{Module Constructions}}{1133}{section.69.2}%
\contentsline {subsection}{\numberline {69.2.1}\textcolor {Chapter }{NaturalGModule}}{1133}{subsection.69.2.1}%
\contentsline {subsection}{\numberline {69.2.2}\textcolor {Chapter }{PermutationGModule}}{1133}{subsection.69.2.2}%
\contentsline {subsection}{\numberline {69.2.3}\textcolor {Chapter }{TrivialGModule}}{1133}{subsection.69.2.3}%
\contentsline {subsection}{\numberline {69.2.4}\textcolor {Chapter }{TensorProductGModule}}{1133}{subsection.69.2.4}%
\contentsline {subsection}{\numberline {69.2.5}\textcolor {Chapter }{DirectSumGModule}}{1133}{subsection.69.2.5}%
\contentsline {subsection}{\numberline {69.2.6}\textcolor {Chapter }{WedgeGModule}}{1133}{subsection.69.2.6}%
\contentsline {section}{\numberline {69.3}\textcolor {Chapter }{Selecting a Different MeatAxe}}{1134}{section.69.3}%
\contentsline {subsection}{\numberline {69.3.1}\textcolor {Chapter }{MTX}}{1134}{subsection.69.3.1}%
\contentsline {section}{\numberline {69.4}\textcolor {Chapter }{Accessing a Module}}{1134}{section.69.4}%
\contentsline {subsection}{\numberline {69.4.1}\textcolor {Chapter }{MTX.Generators}}{1134}{subsection.69.4.1}%
\contentsline {subsection}{\numberline {69.4.2}\textcolor {Chapter }{MTX.Dimension}}{1134}{subsection.69.4.2}%
\contentsline {subsection}{\numberline {69.4.3}\textcolor {Chapter }{MTX.Field}}{1134}{subsection.69.4.3}%
\contentsline {section}{\numberline {69.5}\textcolor {Chapter }{Irreducibility Tests}}{1134}{section.69.5}%
\contentsline {subsection}{\numberline {69.5.1}\textcolor {Chapter }{MTX.IsIrreducible}}{1134}{subsection.69.5.1}%
\contentsline {subsection}{\numberline {69.5.2}\textcolor {Chapter }{MTX.IsAbsolutelyIrreducible}}{1135}{subsection.69.5.2}%
\contentsline {subsection}{\numberline {69.5.3}\textcolor {Chapter }{MTX.DegreeSplittingField}}{1135}{subsection.69.5.3}%
\contentsline {section}{\numberline {69.6}\textcolor {Chapter }{Decomposition of modules}}{1135}{section.69.6}%
\contentsline {subsection}{\numberline {69.6.1}\textcolor {Chapter }{MTX.IsIndecomposable}}{1135}{subsection.69.6.1}%
\contentsline {subsection}{\numberline {69.6.2}\textcolor {Chapter }{MTX.Indecomposition}}{1135}{subsection.69.6.2}%
\contentsline {subsection}{\numberline {69.6.3}\textcolor {Chapter }{MTX.HomogeneousComponents}}{1135}{subsection.69.6.3}%
\contentsline {section}{\numberline {69.7}\textcolor {Chapter }{Finding Submodules}}{1136}{section.69.7}%
\contentsline {subsection}{\numberline {69.7.1}\textcolor {Chapter }{MTX.SubmoduleGModule}}{1136}{subsection.69.7.1}%
\contentsline {subsection}{\numberline {69.7.2}\textcolor {Chapter }{MTX.ProperSubmoduleBasis}}{1136}{subsection.69.7.2}%
\contentsline {subsection}{\numberline {69.7.3}\textcolor {Chapter }{MTX.BasesSubmodules}}{1136}{subsection.69.7.3}%
\contentsline {subsection}{\numberline {69.7.4}\textcolor {Chapter }{MTX.BasesMinimalSubmodules}}{1136}{subsection.69.7.4}%
\contentsline {subsection}{\numberline {69.7.5}\textcolor {Chapter }{MTX.BasesMaximalSubmodules}}{1136}{subsection.69.7.5}%
\contentsline {subsection}{\numberline {69.7.6}\textcolor {Chapter }{MTX.BasisRadical}}{1136}{subsection.69.7.6}%
\contentsline {subsection}{\numberline {69.7.7}\textcolor {Chapter }{MTX.BasisSocle}}{1137}{subsection.69.7.7}%
\contentsline {subsection}{\numberline {69.7.8}\textcolor {Chapter }{MTX.BasesMinimalSupermodules}}{1137}{subsection.69.7.8}%
\contentsline {subsection}{\numberline {69.7.9}\textcolor {Chapter }{MTX.BasesCompositionSeries}}{1137}{subsection.69.7.9}%
\contentsline {subsection}{\numberline {69.7.10}\textcolor {Chapter }{MTX.CompositionFactors}}{1137}{subsection.69.7.10}%
\contentsline {subsection}{\numberline {69.7.11}\textcolor {Chapter }{MTX.CollectedFactors}}{1137}{subsection.69.7.11}%
\contentsline {section}{\numberline {69.8}\textcolor {Chapter }{Induced Actions}}{1137}{section.69.8}%
\contentsline {subsection}{\numberline {69.8.1}\textcolor {Chapter }{MTX.NormedBasisAndBaseChange}}{1137}{subsection.69.8.1}%
\contentsline {subsection}{\numberline {69.8.2}\textcolor {Chapter }{MTX.InducedActionSubmodule}}{1137}{subsection.69.8.2}%
\contentsline {subsection}{\numberline {69.8.3}\textcolor {Chapter }{MTX.InducedActionFactorModule}}{1138}{subsection.69.8.3}%
\contentsline {subsection}{\numberline {69.8.4}\textcolor {Chapter }{MTX.InducedActionSubMatrix}}{1138}{subsection.69.8.4}%
\contentsline {subsection}{\numberline {69.8.5}\textcolor {Chapter }{MTX.InducedAction}}{1138}{subsection.69.8.5}%
\contentsline {section}{\numberline {69.9}\textcolor {Chapter }{Module Homomorphisms}}{1138}{section.69.9}%
\contentsline {subsection}{\numberline {69.9.1}\textcolor {Chapter }{MTX.BasisModuleHomomorphisms}}{1138}{subsection.69.9.1}%
\contentsline {subsection}{\numberline {69.9.2}\textcolor {Chapter }{MTX.BasisModuleEndomorphisms}}{1138}{subsection.69.9.2}%
\contentsline {subsection}{\numberline {69.9.3}\textcolor {Chapter }{MTX.IsomorphismModules}}{1139}{subsection.69.9.3}%
\contentsline {subsection}{\numberline {69.9.4}\textcolor {Chapter }{MTX.ModuleAutomorphisms}}{1139}{subsection.69.9.4}%
\contentsline {section}{\numberline {69.10}\textcolor {Chapter }{Module Homomorphisms for irreducible modules}}{1139}{section.69.10}%
\contentsline {subsection}{\numberline {69.10.1}\textcolor {Chapter }{MTX.IsEquivalent}}{1139}{subsection.69.10.1}%
\contentsline {subsection}{\numberline {69.10.2}\textcolor {Chapter }{MTX.IsomorphismIrred}}{1139}{subsection.69.10.2}%
\contentsline {subsection}{\numberline {69.10.3}\textcolor {Chapter }{MTX.Homomorphism}}{1139}{subsection.69.10.3}%
\contentsline {subsection}{\numberline {69.10.4}\textcolor {Chapter }{MTX.Homomorphisms}}{1140}{subsection.69.10.4}%
\contentsline {subsection}{\numberline {69.10.5}\textcolor {Chapter }{MTX.Distinguish}}{1140}{subsection.69.10.5}%
\contentsline {section}{\numberline {69.11}\textcolor {Chapter }{MeatAxe Functionality for Invariant Forms}}{1140}{section.69.11}%
\contentsline {subsection}{\numberline {69.11.1}\textcolor {Chapter }{MTX.InvariantBilinearForm}}{1140}{subsection.69.11.1}%
\contentsline {subsection}{\numberline {69.11.2}\textcolor {Chapter }{MTX.InvariantSesquilinearForm}}{1140}{subsection.69.11.2}%
\contentsline {subsection}{\numberline {69.11.3}\textcolor {Chapter }{MTX.InvariantQuadraticForm}}{1140}{subsection.69.11.3}%
\contentsline {subsection}{\numberline {69.11.4}\textcolor {Chapter }{MTX.BasisInOrbit}}{1141}{subsection.69.11.4}%
\contentsline {subsection}{\numberline {69.11.5}\textcolor {Chapter }{MTX.OrthogonalSign}}{1141}{subsection.69.11.5}%
\contentsline {section}{\numberline {69.12}\textcolor {Chapter }{The Smash MeatAxe}}{1142}{section.69.12}%
\contentsline {subsection}{\numberline {69.12.1}\textcolor {Chapter }{SMTX.RandomIrreducibleSubGModule}}{1142}{subsection.69.12.1}%
\contentsline {subsection}{\numberline {69.12.2}\textcolor {Chapter }{SMTX.GoodElementGModule}}{1142}{subsection.69.12.2}%
\contentsline {subsection}{\numberline {69.12.3}\textcolor {Chapter }{SMTX.SortHomGModule}}{1142}{subsection.69.12.3}%
\contentsline {subsection}{\numberline {69.12.4}\textcolor {Chapter }{SMTX.MinimalSubGModules}}{1142}{subsection.69.12.4}%
\contentsline {subsection}{\numberline {69.12.5}\textcolor {Chapter }{SMTX.Setter}}{1142}{subsection.69.12.5}%
\contentsline {subsection}{\numberline {69.12.6}\textcolor {Chapter }{SMTX.Getter}}{1142}{subsection.69.12.6}%
\contentsline {subsection}{\numberline {69.12.7}\textcolor {Chapter }{SMTX.IrreducibilityTest}}{1143}{subsection.69.12.7}%
\contentsline {subsection}{\numberline {69.12.8}\textcolor {Chapter }{SMTX.AbsoluteIrreducibilityTest}}{1143}{subsection.69.12.8}%
\contentsline {subsection}{\numberline {69.12.9}\textcolor {Chapter }{SMTX.MinimalSubGModule}}{1143}{subsection.69.12.9}%
\contentsline {subsection}{\numberline {69.12.10}\textcolor {Chapter }{SMTX.MatrixSum}}{1143}{subsection.69.12.10}%
\contentsline {subsection}{\numberline {69.12.11}\textcolor {Chapter }{SMTX.CompleteBasis}}{1143}{subsection.69.12.11}%
\contentsline {section}{\numberline {69.13}\textcolor {Chapter }{Smash MeatAxe Flags}}{1143}{section.69.13}%
\contentsline {subsection}{\numberline {69.13.1}\textcolor {Chapter }{SMTX.Subbasis}}{1143}{subsection.69.13.1}%
\contentsline {subsection}{\numberline {69.13.2}\textcolor {Chapter }{SMTX.AlgEl}}{1144}{subsection.69.13.2}%
\contentsline {subsection}{\numberline {69.13.3}\textcolor {Chapter }{SMTX.AlgElMat}}{1144}{subsection.69.13.3}%
\contentsline {subsection}{\numberline {69.13.4}\textcolor {Chapter }{SMTX.AlgElCharPol}}{1144}{subsection.69.13.4}%
\contentsline {subsection}{\numberline {69.13.5}\textcolor {Chapter }{SMTX.AlgElCharPolFac}}{1144}{subsection.69.13.5}%
\contentsline {subsection}{\numberline {69.13.6}\textcolor {Chapter }{SMTX.AlgElNullspaceVec}}{1144}{subsection.69.13.6}%
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\contentsline {subsection}{\numberline {69.13.8}\textcolor {Chapter }{SMTX.CentMat}}{1144}{subsection.69.13.8}%
\contentsline {subsection}{\numberline {69.13.9}\textcolor {Chapter }{SMTX.CentMatMinPoly}}{1144}{subsection.69.13.9}%
\contentsline {chapter}{\numberline {70}\textcolor {Chapter }{Tables of Marks}}{1145}{chapter.70}%
\contentsline {section}{\numberline {70.1}\textcolor {Chapter }{More about Tables of Marks}}{1145}{section.70.1}%
\contentsline {section}{\numberline {70.2}\textcolor {Chapter }{Table of Marks Objects in GAP}}{1146}{section.70.2}%
\contentsline {section}{\numberline {70.3}\textcolor {Chapter }{Constructing Tables of Marks}}{1147}{section.70.3}%
\contentsline {subsection}{\numberline {70.3.1}\textcolor {Chapter }{TableOfMarks (for a group)}}{1147}{subsection.70.3.1}%
\contentsline {subsection}{\numberline {70.3.2}\textcolor {Chapter }{TableOfMarksByLattice}}{1148}{subsection.70.3.2}%
\contentsline {subsection}{\numberline {70.3.3}\textcolor {Chapter }{LatticeSubgroupsByTom}}{1148}{subsection.70.3.3}%
\contentsline {section}{\numberline {70.4}\textcolor {Chapter }{Printing Tables of Marks}}{1148}{section.70.4}%
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\contentsline {subsection}{\numberline {70.4.2}\textcolor {Chapter }{PrintObj (for a table of marks)}}{1149}{subsection.70.4.2}%
\contentsline {subsection}{\numberline {70.4.3}\textcolor {Chapter }{Display (for a table of marks)}}{1149}{subsection.70.4.3}%
\contentsline {section}{\numberline {70.5}\textcolor {Chapter }{Sorting Tables of Marks}}{1150}{section.70.5}%
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\contentsline {subsection}{\numberline {70.5.2}\textcolor {Chapter }{PermutationTom}}{1151}{subsection.70.5.2}%
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\contentsline {subsection}{\numberline {70.7.2}\textcolor {Chapter }{NrSubsTom}}{1153}{subsection.70.7.2}%
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\contentsline {subsection}{\numberline {70.7.6}\textcolor {Chapter }{FusionsTom}}{1154}{subsection.70.7.6}%
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\contentsline {subsection}{\numberline {70.9.12}\textcolor {Chapter }{MaximalSubgroupsTom}}{1161}{subsection.70.9.12}%
\contentsline {subsection}{\numberline {70.9.13}\textcolor {Chapter }{MinimalSupergroupsTom}}{1162}{subsection.70.9.13}%
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\contentsline {subsection}{\numberline {70.10.2}\textcolor {Chapter }{StraightLineProgramsTom}}{1162}{subsection.70.10.2}%
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\contentsline {section}{\numberline {71.2}\textcolor {Chapter }{History of Character Theory Stuff in GAP}}{1170}{section.71.2}%
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\contentsline {subsection}{\numberline {71.3.2}\textcolor {Chapter }{BrauerTable}}{1172}{subsection.71.3.2}%
\contentsline {subsection}{\numberline {71.3.3}\textcolor {Chapter }{CharacterTableRegular}}{1173}{subsection.71.3.3}%
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\contentsline {subsection}{\numberline {71.9.2}\textcolor {Chapter }{SizesCentralizers}}{1184}{subsection.71.9.2}%
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\contentsline {subsection}{\numberline {71.10.2}\textcolor {Chapter }{ClassPositionsOfAgemo}}{1188}{subsection.71.10.2}%
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\contentsline {subsection}{\numberline {71.10.8}\textcolor {Chapter }{ClassPositionsOfLowerCentralSeries}}{1190}{subsection.71.10.8}%
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\contentsline {subsection}{\numberline {71.11.3}\textcolor {Chapter }{BlocksInfo}}{1193}{subsection.71.11.3}%
\contentsline {subsection}{\numberline {71.11.4}\textcolor {Chapter }{DecompositionMatrix}}{1193}{subsection.71.11.4}%
\contentsline {subsection}{\numberline {71.11.5}\textcolor {Chapter }{LaTeXStringDecompositionMatrix}}{1194}{subsection.71.11.5}%
\contentsline {section}{\numberline {71.12}\textcolor {Chapter }{Other Operations for Character Tables}}{1195}{section.71.12}%
\contentsline {subsection}{\numberline {71.12.1}\textcolor {Chapter }{Index (for two character tables)}}{1195}{subsection.71.12.1}%
\contentsline {subsection}{\numberline {71.12.2}\textcolor {Chapter }{IsInternallyConsistent (for character tables)}}{1195}{subsection.71.12.2}%
\contentsline {subsection}{\numberline {71.12.3}\textcolor {Chapter }{IsPSolvableCharacterTable}}{1196}{subsection.71.12.3}%
\contentsline {subsection}{\numberline {71.12.4}\textcolor {Chapter }{IsClassFusionOfNormalSubgroup}}{1196}{subsection.71.12.4}%
\contentsline {subsection}{\numberline {71.12.5}\textcolor {Chapter }{Indicator}}{1197}{subsection.71.12.5}%
\contentsline {subsection}{\numberline {71.12.6}\textcolor {Chapter }{NrPolyhedralSubgroups}}{1197}{subsection.71.12.6}%
\contentsline {subsection}{\numberline {71.12.7}\textcolor {Chapter }{ClassMultiplicationCoefficient (for character tables)}}{1198}{subsection.71.12.7}%
\contentsline {subsection}{\numberline {71.12.8}\textcolor {Chapter }{ClassStructureCharTable}}{1198}{subsection.71.12.8}%
\contentsline {subsection}{\numberline {71.12.9}\textcolor {Chapter }{MatClassMultCoeffsCharTable}}{1199}{subsection.71.12.9}%
\contentsline {section}{\numberline {71.13}\textcolor {Chapter }{Printing Character Tables}}{1199}{section.71.13}%
\contentsline {subsection}{\numberline {71.13.1}\textcolor {Chapter }{ViewObj (for a character table)}}{1199}{subsection.71.13.1}%
\contentsline {subsection}{\numberline {71.13.2}\textcolor {Chapter }{PrintObj (for a character table)}}{1199}{subsection.71.13.2}%
\contentsline {subsection}{\numberline {71.13.3}\textcolor {Chapter }{Display (for a character table)}}{1199}{subsection.71.13.3}%
\contentsline {subsection}{\numberline {71.13.4}\textcolor {Chapter }{DisplayOptions}}{1201}{subsection.71.13.4}%
\contentsline {subsection}{\numberline {71.13.5}\textcolor {Chapter }{PrintCharacterTable}}{1203}{subsection.71.13.5}%
\contentsline {section}{\numberline {71.14}\textcolor {Chapter }{Computing the Irreducible Characters of a Group}}{1203}{section.71.14}%
\contentsline {subsection}{\numberline {71.14.1}\textcolor {Chapter }{IrrDixonSchneider}}{1204}{subsection.71.14.1}%
\contentsline {subsection}{\numberline {71.14.2}\textcolor {Chapter }{IrrConlon}}{1204}{subsection.71.14.2}%
\contentsline {subsection}{\numberline {71.14.3}\textcolor {Chapter }{IrrBaumClausen}}{1204}{subsection.71.14.3}%
\contentsline {subsection}{\numberline {71.14.4}\textcolor {Chapter }{IrreducibleRepresentations}}{1205}{subsection.71.14.4}%
\contentsline {subsection}{\numberline {71.14.5}\textcolor {Chapter }{IrreducibleRepresentationsDixon}}{1206}{subsection.71.14.5}%
\contentsline {section}{\numberline {71.15}\textcolor {Chapter }{Representations Given by Modules}}{1206}{section.71.15}%
\contentsline {subsection}{\numberline {71.15.1}\textcolor {Chapter }{IrreducibleModules}}{1206}{subsection.71.15.1}%
\contentsline {subsection}{\numberline {71.15.2}\textcolor {Chapter }{AbsolutelyIrreducibleModules}}{1207}{subsection.71.15.2}%
\contentsline {subsection}{\numberline {71.15.3}\textcolor {Chapter }{RegularModule}}{1207}{subsection.71.15.3}%
\contentsline {section}{\numberline {71.16}\textcolor {Chapter }{The Dixon\texttt {\symbol {45}}Schneider Algorithm}}{1207}{section.71.16}%
\contentsline {section}{\numberline {71.17}\textcolor {Chapter }{Advanced Methods for Dixon\texttt {\symbol {45}}Schneider Calculations}}{1208}{section.71.17}%
\contentsline {subsection}{\numberline {71.17.1}\textcolor {Chapter }{DixonRecord}}{1208}{subsection.71.17.1}%
\contentsline {subsection}{\numberline {71.17.2}\textcolor {Chapter }{DixonInit}}{1208}{subsection.71.17.2}%
\contentsline {subsection}{\numberline {71.17.3}\textcolor {Chapter }{DixontinI}}{1208}{subsection.71.17.3}%
\contentsline {subsection}{\numberline {71.17.4}\textcolor {Chapter }{DixonSplit}}{1208}{subsection.71.17.4}%
\contentsline {subsection}{\numberline {71.17.5}\textcolor {Chapter }{BestSplittingMatrix}}{1209}{subsection.71.17.5}%
\contentsline {subsection}{\numberline {71.17.6}\textcolor {Chapter }{DxIncludeIrreducibles}}{1209}{subsection.71.17.6}%
\contentsline {subsection}{\numberline {71.17.7}\textcolor {Chapter }{SplitCharacters}}{1209}{subsection.71.17.7}%
\contentsline {subsection}{\numberline {71.17.8}\textcolor {Chapter }{IsDxLargeGroup}}{1209}{subsection.71.17.8}%
\contentsline {section}{\numberline {71.18}\textcolor {Chapter }{Components of a Dixon Record}}{1210}{section.71.18}%
\contentsline {section}{\numberline {71.19}\textcolor {Chapter }{An Example of Advanced Dixon\texttt {\symbol {45}}Schneider Calculations}}{1210}{section.71.19}%
\contentsline {section}{\numberline {71.20}\textcolor {Chapter }{Constructing Character Tables from Others}}{1211}{section.71.20}%
\contentsline {subsection}{\numberline {71.20.1}\textcolor {Chapter }{CharacterTableDirectProduct}}{1212}{subsection.71.20.1}%
\contentsline {subsection}{\numberline {71.20.2}\textcolor {Chapter }{FactorsOfDirectProduct}}{1213}{subsection.71.20.2}%
\contentsline {subsection}{\numberline {71.20.3}\textcolor {Chapter }{CharacterTableFactorGroup}}{1213}{subsection.71.20.3}%
\contentsline {subsection}{\numberline {71.20.4}\textcolor {Chapter }{CharacterTableIsoclinic}}{1213}{subsection.71.20.4}%
\contentsline {subsection}{\numberline {71.20.5}\textcolor {Chapter }{CharacterTableOfNormalSubgroup}}{1216}{subsection.71.20.5}%
\contentsline {subsection}{\numberline {71.20.6}\textcolor {Chapter }{CharacterTableWreathSymmetric}}{1217}{subsection.71.20.6}%
\contentsline {subsection}{\numberline {71.20.7}\textcolor {Chapter }{CharacterValueWreathSymmetric}}{1217}{subsection.71.20.7}%
\contentsline {section}{\numberline {71.21}\textcolor {Chapter }{Sorted Character Tables}}{1218}{section.71.21}%
\contentsline {subsection}{\numberline {71.21.1}\textcolor {Chapter }{CharacterTableWithSortedCharacters}}{1218}{subsection.71.21.1}%
\contentsline {subsection}{\numberline {71.21.2}\textcolor {Chapter }{SortedCharacters}}{1218}{subsection.71.21.2}%
\contentsline {subsection}{\numberline {71.21.3}\textcolor {Chapter }{CharacterTableWithSortedClasses}}{1219}{subsection.71.21.3}%
\contentsline {subsection}{\numberline {71.21.4}\textcolor {Chapter }{SortedCharacterTable (w.r.t. a normal subgroup)}}{1219}{subsection.71.21.4}%
\contentsline {subsection}{\numberline {71.21.5}\textcolor {Chapter }{ClassPermutation}}{1220}{subsection.71.21.5}%
\contentsline {section}{\numberline {71.22}\textcolor {Chapter }{Automorphisms and Equivalence of Character Tables}}{1221}{section.71.22}%
\contentsline {subsection}{\numberline {71.22.1}\textcolor {Chapter }{MatrixAutomorphisms}}{1221}{subsection.71.22.1}%
\contentsline {subsection}{\numberline {71.22.2}\textcolor {Chapter }{TableAutomorphisms}}{1221}{subsection.71.22.2}%
\contentsline {subsection}{\numberline {71.22.3}\textcolor {Chapter }{TransformingPermutations}}{1222}{subsection.71.22.3}%
\contentsline {subsection}{\numberline {71.22.4}\textcolor {Chapter }{TransformingPermutationsCharacterTables}}{1222}{subsection.71.22.4}%
\contentsline {subsection}{\numberline {71.22.5}\textcolor {Chapter }{FamiliesOfRows}}{1223}{subsection.71.22.5}%
\contentsline {section}{\numberline {71.23}\textcolor {Chapter }{Storing Normal Subgroup Information}}{1223}{section.71.23}%
\contentsline {subsection}{\numberline {71.23.1}\textcolor {Chapter }{NormalSubgroupClassesInfo}}{1223}{subsection.71.23.1}%
\contentsline {subsection}{\numberline {71.23.2}\textcolor {Chapter }{ClassPositionsOfNormalSubgroup}}{1224}{subsection.71.23.2}%
\contentsline {subsection}{\numberline {71.23.3}\textcolor {Chapter }{NormalSubgroupClasses}}{1224}{subsection.71.23.3}%
\contentsline {subsection}{\numberline {71.23.4}\textcolor {Chapter }{FactorGroupNormalSubgroupClasses}}{1224}{subsection.71.23.4}%
\contentsline {chapter}{\numberline {72}\textcolor {Chapter }{Class Functions}}{1226}{chapter.72}%
\contentsline {section}{\numberline {72.1}\textcolor {Chapter }{Why Class Functions?}}{1226}{section.72.1}%
\contentsline {subsection}{\numberline {72.1.1}\textcolor {Chapter }{IsClassFunction}}{1228}{subsection.72.1.1}%
\contentsline {section}{\numberline {72.2}\textcolor {Chapter }{Basic Operations for Class Functions}}{1229}{section.72.2}%
\contentsline {subsection}{\numberline {72.2.1}\textcolor {Chapter }{UnderlyingCharacterTable}}{1229}{subsection.72.2.1}%
\contentsline {subsection}{\numberline {72.2.2}\textcolor {Chapter }{ValuesOfClassFunction}}{1229}{subsection.72.2.2}%
\contentsline {section}{\numberline {72.3}\textcolor {Chapter }{Comparison of Class Functions}}{1230}{section.72.3}%
\contentsline {section}{\numberline {72.4}\textcolor {Chapter }{Arithmetic Operations for Class Functions}}{1230}{section.72.4}%
\contentsline {subsection}{\numberline {72.4.1}\textcolor {Chapter }{Characteristic (for a class function)}}{1233}{subsection.72.4.1}%
\contentsline {subsection}{\numberline {72.4.2}\textcolor {Chapter }{ComplexConjugate (for a class function)}}{1233}{subsection.72.4.2}%
\contentsline {subsection}{\numberline {72.4.3}\textcolor {Chapter }{Order (for a class function)}}{1234}{subsection.72.4.3}%
\contentsline {section}{\numberline {72.5}\textcolor {Chapter }{Printing Class Functions}}{1234}{section.72.5}%
\contentsline {subsection}{\numberline {72.5.1}\textcolor {Chapter }{ViewObj (for class functions)}}{1234}{subsection.72.5.1}%
\contentsline {subsection}{\numberline {72.5.2}\textcolor {Chapter }{PrintObj (for class functions)}}{1234}{subsection.72.5.2}%
\contentsline {subsection}{\numberline {72.5.3}\textcolor {Chapter }{Display (for class functions)}}{1234}{subsection.72.5.3}%
\contentsline {section}{\numberline {72.6}\textcolor {Chapter }{Creating Class Functions from Values Lists}}{1235}{section.72.6}%
\contentsline {subsection}{\numberline {72.6.1}\textcolor {Chapter }{ClassFunction (for a character table and a list)}}{1235}{subsection.72.6.1}%
\contentsline {subsection}{\numberline {72.6.2}\textcolor {Chapter }{VirtualCharacter (for a character table and a list)}}{1235}{subsection.72.6.2}%
\contentsline {subsection}{\numberline {72.6.3}\textcolor {Chapter }{Character (for a character table and a list)}}{1235}{subsection.72.6.3}%
\contentsline {subsection}{\numberline {72.6.4}\textcolor {Chapter }{ClassFunctionSameType}}{1236}{subsection.72.6.4}%
\contentsline {section}{\numberline {72.7}\textcolor {Chapter }{Creating Class Functions using Groups}}{1236}{section.72.7}%
\contentsline {subsection}{\numberline {72.7.1}\textcolor {Chapter }{TrivialCharacter}}{1236}{subsection.72.7.1}%
\contentsline {subsection}{\numberline {72.7.2}\textcolor {Chapter }{NaturalCharacter (for a group)}}{1236}{subsection.72.7.2}%
\contentsline {subsection}{\numberline {72.7.3}\textcolor {Chapter }{PermutationCharacter}}{1237}{subsection.72.7.3}%
\contentsline {section}{\numberline {72.8}\textcolor {Chapter }{Operations for Class Functions}}{1237}{section.72.8}%
\contentsline {subsection}{\numberline {72.8.1}\textcolor {Chapter }{IsCharacter}}{1238}{subsection.72.8.1}%
\contentsline {subsection}{\numberline {72.8.2}\textcolor {Chapter }{IsVirtualCharacter}}{1238}{subsection.72.8.2}%
\contentsline {subsection}{\numberline {72.8.3}\textcolor {Chapter }{IsIrreducibleCharacter}}{1238}{subsection.72.8.3}%
\contentsline {subsection}{\numberline {72.8.4}\textcolor {Chapter }{DegreeOfCharacter}}{1238}{subsection.72.8.4}%
\contentsline {subsection}{\numberline {72.8.5}\textcolor {Chapter }{ScalarProduct (for characters)}}{1239}{subsection.72.8.5}%
\contentsline {subsection}{\numberline {72.8.6}\textcolor {Chapter }{MatScalarProducts}}{1239}{subsection.72.8.6}%
\contentsline {subsection}{\numberline {72.8.7}\textcolor {Chapter }{Norm (for a class function)}}{1239}{subsection.72.8.7}%
\contentsline {subsection}{\numberline {72.8.8}\textcolor {Chapter }{ConstituentsOfCharacter}}{1240}{subsection.72.8.8}%
\contentsline {subsection}{\numberline {72.8.9}\textcolor {Chapter }{KernelOfCharacter}}{1240}{subsection.72.8.9}%
\contentsline {subsection}{\numberline {72.8.10}\textcolor {Chapter }{ClassPositionsOfKernel}}{1240}{subsection.72.8.10}%
\contentsline {subsection}{\numberline {72.8.11}\textcolor {Chapter }{CentreOfCharacter}}{1241}{subsection.72.8.11}%
\contentsline {subsection}{\numberline {72.8.12}\textcolor {Chapter }{ClassPositionsOfCentre (for a character)}}{1241}{subsection.72.8.12}%
\contentsline {subsection}{\numberline {72.8.13}\textcolor {Chapter }{InertiaSubgroup}}{1241}{subsection.72.8.13}%
\contentsline {subsection}{\numberline {72.8.14}\textcolor {Chapter }{CycleStructureClass}}{1242}{subsection.72.8.14}%
\contentsline {subsection}{\numberline {72.8.15}\textcolor {Chapter }{IsTransitive (for a character)}}{1242}{subsection.72.8.15}%
\contentsline {subsection}{\numberline {72.8.16}\textcolor {Chapter }{Transitivity (for a character)}}{1242}{subsection.72.8.16}%
\contentsline {subsection}{\numberline {72.8.17}\textcolor {Chapter }{CentralCharacter}}{1242}{subsection.72.8.17}%
\contentsline {subsection}{\numberline {72.8.18}\textcolor {Chapter }{DeterminantOfCharacter}}{1242}{subsection.72.8.18}%
\contentsline {subsection}{\numberline {72.8.19}\textcolor {Chapter }{EigenvaluesChar}}{1243}{subsection.72.8.19}%
\contentsline {subsection}{\numberline {72.8.20}\textcolor {Chapter }{Tensored}}{1243}{subsection.72.8.20}%
\contentsline {subsection}{\numberline {72.8.21}\textcolor {Chapter }{TensorProduct (for characters)}}{1244}{subsection.72.8.21}%
\contentsline {section}{\numberline {72.9}\textcolor {Chapter }{Restricted and Induced Class Functions}}{1244}{section.72.9}%
\contentsline {subsection}{\numberline {72.9.1}\textcolor {Chapter }{RestrictedClassFunction}}{1244}{subsection.72.9.1}%
\contentsline {subsection}{\numberline {72.9.2}\textcolor {Chapter }{RestrictedClassFunctions}}{1245}{subsection.72.9.2}%
\contentsline {subsection}{\numberline {72.9.3}\textcolor {Chapter }{InducedClassFunction}}{1245}{subsection.72.9.3}%
\contentsline {subsection}{\numberline {72.9.4}\textcolor {Chapter }{InducedClassFunctions}}{1245}{subsection.72.9.4}%
\contentsline {subsection}{\numberline {72.9.5}\textcolor {Chapter }{InducedClassFunctionsByFusionMap}}{1246}{subsection.72.9.5}%
\contentsline {subsection}{\numberline {72.9.6}\textcolor {Chapter }{InducedCyclic}}{1246}{subsection.72.9.6}%
\contentsline {section}{\numberline {72.10}\textcolor {Chapter }{Reducing Virtual Characters}}{1246}{section.72.10}%
\contentsline {subsection}{\numberline {72.10.1}\textcolor {Chapter }{ReducedClassFunctions}}{1247}{subsection.72.10.1}%
\contentsline {subsection}{\numberline {72.10.2}\textcolor {Chapter }{ReducedCharacters}}{1247}{subsection.72.10.2}%
\contentsline {subsection}{\numberline {72.10.3}\textcolor {Chapter }{IrreducibleDifferences}}{1247}{subsection.72.10.3}%
\contentsline {subsection}{\numberline {72.10.4}\textcolor {Chapter }{LLL}}{1248}{subsection.72.10.4}%
\contentsline {subsection}{\numberline {72.10.5}\textcolor {Chapter }{Extract}}{1249}{subsection.72.10.5}%
\contentsline {subsection}{\numberline {72.10.6}\textcolor {Chapter }{OrthogonalEmbeddingsSpecialDimension}}{1250}{subsection.72.10.6}%
\contentsline {subsection}{\numberline {72.10.7}\textcolor {Chapter }{Decreased}}{1251}{subsection.72.10.7}%
\contentsline {subsection}{\numberline {72.10.8}\textcolor {Chapter }{DnLattice}}{1252}{subsection.72.10.8}%
\contentsline {subsection}{\numberline {72.10.9}\textcolor {Chapter }{DnLatticeIterative}}{1253}{subsection.72.10.9}%
\contentsline {section}{\numberline {72.11}\textcolor {Chapter }{Symmetrizations of Class Functions}}{1253}{section.72.11}%
\contentsline {subsection}{\numberline {72.11.1}\textcolor {Chapter }{Symmetrizations}}{1253}{subsection.72.11.1}%
\contentsline {subsection}{\numberline {72.11.2}\textcolor {Chapter }{SymmetricParts}}{1254}{subsection.72.11.2}%
\contentsline {subsection}{\numberline {72.11.3}\textcolor {Chapter }{AntiSymmetricParts}}{1254}{subsection.72.11.3}%
\contentsline {subsection}{\numberline {72.11.4}\textcolor {Chapter }{ExteriorPower (for a character)}}{1255}{subsection.72.11.4}%
\contentsline {subsection}{\numberline {72.11.5}\textcolor {Chapter }{SymmetricPower (for a character)}}{1255}{subsection.72.11.5}%
\contentsline {subsection}{\numberline {72.11.6}\textcolor {Chapter }{OrthogonalComponents}}{1255}{subsection.72.11.6}%
\contentsline {subsection}{\numberline {72.11.7}\textcolor {Chapter }{SymplecticComponents}}{1256}{subsection.72.11.7}%
\contentsline {section}{\numberline {72.12}\textcolor {Chapter }{Molien Series}}{1257}{section.72.12}%
\contentsline {subsection}{\numberline {72.12.1}\textcolor {Chapter }{MolienSeries}}{1257}{subsection.72.12.1}%
\contentsline {subsection}{\numberline {72.12.2}\textcolor {Chapter }{MolienSeriesInfo}}{1257}{subsection.72.12.2}%
\contentsline {subsection}{\numberline {72.12.3}\textcolor {Chapter }{ValueMolienSeries}}{1258}{subsection.72.12.3}%
\contentsline {subsection}{\numberline {72.12.4}\textcolor {Chapter }{MolienSeriesWithGivenDenominator}}{1259}{subsection.72.12.4}%
\contentsline {section}{\numberline {72.13}\textcolor {Chapter }{Possible Permutation Characters}}{1259}{section.72.13}%
\contentsline {subsection}{\numberline {72.13.1}\textcolor {Chapter }{PermCharInfo}}{1260}{subsection.72.13.1}%
\contentsline {subsection}{\numberline {72.13.2}\textcolor {Chapter }{PermCharInfoRelative}}{1261}{subsection.72.13.2}%
\contentsline {section}{\numberline {72.14}\textcolor {Chapter }{Computing Possible Permutation Characters}}{1262}{section.72.14}%
\contentsline {subsection}{\numberline {72.14.1}\textcolor {Chapter }{PermChars}}{1262}{subsection.72.14.1}%
\contentsline {subsection}{\numberline {72.14.2}\textcolor {Chapter }{TestPerm1, ..., TestPerm5}}{1264}{subsection.72.14.2}%
\contentsline {subsection}{\numberline {72.14.3}\textcolor {Chapter }{PermBounds}}{1265}{subsection.72.14.3}%
\contentsline {subsection}{\numberline {72.14.4}\textcolor {Chapter }{PermComb}}{1266}{subsection.72.14.4}%
\contentsline {subsection}{\numberline {72.14.5}\textcolor {Chapter }{Inequalities}}{1266}{subsection.72.14.5}%
\contentsline {section}{\numberline {72.15}\textcolor {Chapter }{Operations for Brauer Characters}}{1267}{section.72.15}%
\contentsline {subsection}{\numberline {72.15.1}\textcolor {Chapter }{FrobeniusCharacterValue}}{1267}{subsection.72.15.1}%
\contentsline {subsection}{\numberline {72.15.2}\textcolor {Chapter }{BrauerCharacterValue}}{1267}{subsection.72.15.2}%
\contentsline {subsection}{\numberline {72.15.3}\textcolor {Chapter }{SizeOfFieldOfDefinition}}{1268}{subsection.72.15.3}%
\contentsline {subsection}{\numberline {72.15.4}\textcolor {Chapter }{RealizableBrauerCharacters}}{1268}{subsection.72.15.4}%
\contentsline {section}{\numberline {72.16}\textcolor {Chapter }{Domains Generated by Class Functions}}{1268}{section.72.16}%
\contentsline {chapter}{\numberline {73}\textcolor {Chapter }{Maps Concerning Character Tables}}{1269}{chapter.73}%
\contentsline {section}{\numberline {73.1}\textcolor {Chapter }{Power Maps}}{1269}{section.73.1}%
\contentsline {subsection}{\numberline {73.1.1}\textcolor {Chapter }{PowerMap}}{1270}{subsection.73.1.1}%
\contentsline {subsection}{\numberline {73.1.2}\textcolor {Chapter }{PossiblePowerMaps}}{1271}{subsection.73.1.2}%
\contentsline {subsection}{\numberline {73.1.3}\textcolor {Chapter }{ElementOrdersPowerMap}}{1272}{subsection.73.1.3}%
\contentsline {subsection}{\numberline {73.1.4}\textcolor {Chapter }{PowerMapByComposition}}{1272}{subsection.73.1.4}%
\contentsline {section}{\numberline {73.2}\textcolor {Chapter }{Orbits on Sets of Possible Power Maps}}{1273}{section.73.2}%
\contentsline {subsection}{\numberline {73.2.1}\textcolor {Chapter }{OrbitPowerMaps}}{1273}{subsection.73.2.1}%
\contentsline {subsection}{\numberline {73.2.2}\textcolor {Chapter }{RepresentativesPowerMaps}}{1273}{subsection.73.2.2}%
\contentsline {section}{\numberline {73.3}\textcolor {Chapter }{Class Fusions between Character Tables}}{1274}{section.73.3}%
\contentsline {subsection}{\numberline {73.3.1}\textcolor {Chapter }{FusionConjugacyClasses}}{1275}{subsection.73.3.1}%
\contentsline {subsection}{\numberline {73.3.2}\textcolor {Chapter }{ComputedClassFusions}}{1276}{subsection.73.3.2}%
\contentsline {subsection}{\numberline {73.3.3}\textcolor {Chapter }{GetFusionMap}}{1276}{subsection.73.3.3}%
\contentsline {subsection}{\numberline {73.3.4}\textcolor {Chapter }{StoreFusion}}{1277}{subsection.73.3.4}%
\contentsline {subsection}{\numberline {73.3.5}\textcolor {Chapter }{NamesOfFusionSources}}{1277}{subsection.73.3.5}%
\contentsline {subsection}{\numberline {73.3.6}\textcolor {Chapter }{PossibleClassFusions}}{1278}{subsection.73.3.6}%
\contentsline {subsection}{\numberline {73.3.7}\textcolor {Chapter }{ConsiderStructureConstants}}{1279}{subsection.73.3.7}%
\contentsline {section}{\numberline {73.4}\textcolor {Chapter }{Orbits on Sets of Possible Class Fusions}}{1280}{section.73.4}%
\contentsline {subsection}{\numberline {73.4.1}\textcolor {Chapter }{OrbitFusions}}{1280}{subsection.73.4.1}%
\contentsline {subsection}{\numberline {73.4.2}\textcolor {Chapter }{RepresentativesFusions}}{1280}{subsection.73.4.2}%
\contentsline {section}{\numberline {73.5}\textcolor {Chapter }{Parametrized Maps}}{1280}{section.73.5}%
\contentsline {subsection}{\numberline {73.5.1}\textcolor {Chapter }{CompositionMaps}}{1281}{subsection.73.5.1}%
\contentsline {subsection}{\numberline {73.5.2}\textcolor {Chapter }{InverseMap}}{1281}{subsection.73.5.2}%
\contentsline {subsection}{\numberline {73.5.3}\textcolor {Chapter }{ProjectionMap}}{1282}{subsection.73.5.3}%
\contentsline {subsection}{\numberline {73.5.4}\textcolor {Chapter }{Indirected}}{1282}{subsection.73.5.4}%
\contentsline {subsection}{\numberline {73.5.5}\textcolor {Chapter }{Parametrized}}{1282}{subsection.73.5.5}%
\contentsline {subsection}{\numberline {73.5.6}\textcolor {Chapter }{ContainedMaps}}{1283}{subsection.73.5.6}%
\contentsline {subsection}{\numberline {73.5.7}\textcolor {Chapter }{UpdateMap}}{1283}{subsection.73.5.7}%
\contentsline {subsection}{\numberline {73.5.8}\textcolor {Chapter }{MeetMaps}}{1284}{subsection.73.5.8}%
\contentsline {subsection}{\numberline {73.5.9}\textcolor {Chapter }{CommutativeDiagram}}{1284}{subsection.73.5.9}%
\contentsline {subsection}{\numberline {73.5.10}\textcolor {Chapter }{CheckFixedPoints}}{1285}{subsection.73.5.10}%
\contentsline {subsection}{\numberline {73.5.11}\textcolor {Chapter }{TransferDiagram}}{1285}{subsection.73.5.11}%
\contentsline {subsection}{\numberline {73.5.12}\textcolor {Chapter }{TestConsistencyMaps}}{1286}{subsection.73.5.12}%
\contentsline {subsection}{\numberline {73.5.13}\textcolor {Chapter }{Indeterminateness}}{1286}{subsection.73.5.13}%
\contentsline {subsection}{\numberline {73.5.14}\textcolor {Chapter }{PrintAmbiguity}}{1287}{subsection.73.5.14}%
\contentsline {subsection}{\numberline {73.5.15}\textcolor {Chapter }{ContainedSpecialVectors}}{1287}{subsection.73.5.15}%
\contentsline {subsection}{\numberline {73.5.16}\textcolor {Chapter }{CollapsedMat}}{1288}{subsection.73.5.16}%
\contentsline {subsection}{\numberline {73.5.17}\textcolor {Chapter }{ContainedDecomposables}}{1289}{subsection.73.5.17}%
\contentsline {section}{\numberline {73.6}\textcolor {Chapter }{Subroutines for the Construction of Power Maps}}{1289}{section.73.6}%
\contentsline {subsection}{\numberline {73.6.1}\textcolor {Chapter }{InitPowerMap}}{1290}{subsection.73.6.1}%
\contentsline {subsection}{\numberline {73.6.2}\textcolor {Chapter }{Congruences (for character tables)}}{1290}{subsection.73.6.2}%
\contentsline {subsection}{\numberline {73.6.3}\textcolor {Chapter }{ConsiderKernels}}{1291}{subsection.73.6.3}%
\contentsline {subsection}{\numberline {73.6.4}\textcolor {Chapter }{ConsiderSmallerPowerMaps}}{1291}{subsection.73.6.4}%
\contentsline {subsection}{\numberline {73.6.5}\textcolor {Chapter }{MinusCharacter}}{1291}{subsection.73.6.5}%
\contentsline {subsection}{\numberline {73.6.6}\textcolor {Chapter }{PowerMapsAllowedBySymmetrizations}}{1292}{subsection.73.6.6}%
\contentsline {section}{\numberline {73.7}\textcolor {Chapter }{Subroutines for the Construction of Class Fusions}}{1293}{section.73.7}%
\contentsline {subsection}{\numberline {73.7.1}\textcolor {Chapter }{InitFusion}}{1293}{subsection.73.7.1}%
\contentsline {subsection}{\numberline {73.7.2}\textcolor {Chapter }{CheckPermChar}}{1293}{subsection.73.7.2}%
\contentsline {subsection}{\numberline {73.7.3}\textcolor {Chapter }{ConsiderTableAutomorphisms}}{1294}{subsection.73.7.3}%
\contentsline {subsection}{\numberline {73.7.4}\textcolor {Chapter }{FusionsAllowedByRestrictions}}{1294}{subsection.73.7.4}%
\contentsline {chapter}{\numberline {74}\textcolor {Chapter }{Unknowns}}{1296}{chapter.74}%
\contentsline {section}{\numberline {74.1}\textcolor {Chapter }{More about Unknowns}}{1296}{section.74.1}%
\contentsline {subsection}{\numberline {74.1.1}\textcolor {Chapter }{Unknown}}{1296}{subsection.74.1.1}%
\contentsline {subsection}{\numberline {74.1.2}\textcolor {Chapter }{LargestUnknown}}{1297}{subsection.74.1.2}%
\contentsline {subsection}{\numberline {74.1.3}\textcolor {Chapter }{IsUnknown}}{1297}{subsection.74.1.3}%
\contentsline {subsection}{\numberline {74.1.4}\textcolor {Chapter }{Comparison of Unknowns}}{1297}{subsection.74.1.4}%
\contentsline {subsection}{\numberline {74.1.5}\textcolor {Chapter }{Arithmetical Operations for Unknowns}}{1297}{subsection.74.1.5}%
\contentsline {chapter}{\numberline {75}\textcolor {Chapter }{Monomiality Questions}}{1299}{chapter.75}%
\contentsline {section}{\numberline {75.1}\textcolor {Chapter }{InfoMonomial (Info Class)}}{1300}{section.75.1}%
\contentsline {subsection}{\numberline {75.1.1}\textcolor {Chapter }{InfoMonomial}}{1300}{subsection.75.1.1}%
\contentsline {section}{\numberline {75.2}\textcolor {Chapter }{Character Degrees and Derived Length}}{1300}{section.75.2}%
\contentsline {subsection}{\numberline {75.2.1}\textcolor {Chapter }{Alpha}}{1300}{subsection.75.2.1}%
\contentsline {subsection}{\numberline {75.2.2}\textcolor {Chapter }{Delta}}{1300}{subsection.75.2.2}%
\contentsline {subsection}{\numberline {75.2.3}\textcolor {Chapter }{IsBergerCondition}}{1300}{subsection.75.2.3}%
\contentsline {section}{\numberline {75.3}\textcolor {Chapter }{Primitivity of Characters}}{1301}{section.75.3}%
\contentsline {subsection}{\numberline {75.3.1}\textcolor {Chapter }{TestHomogeneous}}{1301}{subsection.75.3.1}%
\contentsline {subsection}{\numberline {75.3.2}\textcolor {Chapter }{IsPrimitiveCharacter}}{1302}{subsection.75.3.2}%
\contentsline {subsection}{\numberline {75.3.3}\textcolor {Chapter }{TestQuasiPrimitive}}{1302}{subsection.75.3.3}%
\contentsline {subsection}{\numberline {75.3.4}\textcolor {Chapter }{TestInducedFromNormalSubgroup}}{1303}{subsection.75.3.4}%
\contentsline {section}{\numberline {75.4}\textcolor {Chapter }{Testing Monomiality}}{1303}{section.75.4}%
\contentsline {subsection}{\numberline {75.4.1}\textcolor {Chapter }{TestMonomial}}{1303}{subsection.75.4.1}%
\contentsline {subsection}{\numberline {75.4.2}\textcolor {Chapter }{TestMonomialUseLattice}}{1304}{subsection.75.4.2}%
\contentsline {subsection}{\numberline {75.4.3}\textcolor {Chapter }{IsMonomialNumber}}{1304}{subsection.75.4.3}%
\contentsline {subsection}{\numberline {75.4.4}\textcolor {Chapter }{TestMonomialQuick}}{1305}{subsection.75.4.4}%
\contentsline {subsection}{\numberline {75.4.5}\textcolor {Chapter }{TestSubnormallyMonomial}}{1305}{subsection.75.4.5}%
\contentsline {subsection}{\numberline {75.4.6}\textcolor {Chapter }{TestRelativelySM}}{1306}{subsection.75.4.6}%
\contentsline {section}{\numberline {75.5}\textcolor {Chapter }{Minimal Nonmonomial Groups}}{1307}{section.75.5}%
\contentsline {subsection}{\numberline {75.5.1}\textcolor {Chapter }{IsMinimalNonmonomial}}{1307}{subsection.75.5.1}%
\contentsline {subsection}{\numberline {75.5.2}\textcolor {Chapter }{MinimalNonmonomialGroup}}{1307}{subsection.75.5.2}%
\contentsline {chapter}{\numberline {76}\textcolor {Chapter }{Using and Developing \textsf {GAP} Packages}}{1308}{chapter.76}%
\contentsline {section}{\numberline {76.1}\textcolor {Chapter }{Installing a \textsf {GAP} Package}}{1308}{section.76.1}%
\contentsline {section}{\numberline {76.2}\textcolor {Chapter }{Loading a \textsf {GAP} Package}}{1309}{section.76.2}%
\contentsline {subsection}{\numberline {76.2.1}\textcolor {Chapter }{LoadPackage}}{1309}{subsection.76.2.1}%
\contentsline {subsection}{\numberline {76.2.2}\textcolor {Chapter }{Automatic loading of \textsf {GAP} packages}}{1310}{subsection.76.2.2}%
\contentsline {subsection}{\numberline {76.2.3}\textcolor {Chapter }{SetPackagePath}}{1310}{subsection.76.2.3}%
\contentsline {subsection}{\numberline {76.2.4}\textcolor {Chapter }{ExtendRootDirectories}}{1311}{subsection.76.2.4}%
\contentsline {subsection}{\numberline {76.2.5}\textcolor {Chapter }{ExtendPackageDirectories}}{1311}{subsection.76.2.5}%
\contentsline {subsection}{\numberline {76.2.6}\textcolor {Chapter }{DisplayPackageLoadingLog}}{1311}{subsection.76.2.6}%
\contentsline {section}{\numberline {76.3}\textcolor {Chapter }{Functions for \textsf {GAP} Packages}}{1312}{section.76.3}%
\contentsline {subsection}{\numberline {76.3.1}\textcolor {Chapter }{ReadPackage}}{1312}{subsection.76.3.1}%
\contentsline {subsection}{\numberline {76.3.2}\textcolor {Chapter }{TestPackageAvailability}}{1313}{subsection.76.3.2}%
\contentsline {subsection}{\numberline {76.3.3}\textcolor {Chapter }{IsPackageLoaded}}{1313}{subsection.76.3.3}%
\contentsline {subsection}{\numberline {76.3.4}\textcolor {Chapter }{IsPackageMarkedForLoading}}{1313}{subsection.76.3.4}%
\contentsline {subsection}{\numberline {76.3.5}\textcolor {Chapter }{TestPackage}}{1314}{subsection.76.3.5}%
\contentsline {subsection}{\numberline {76.3.6}\textcolor {Chapter }{InstalledPackageVersion}}{1314}{subsection.76.3.6}%
\contentsline {subsection}{\numberline {76.3.7}\textcolor {Chapter }{DirectoriesPackageLibrary}}{1314}{subsection.76.3.7}%
\contentsline {subsection}{\numberline {76.3.8}\textcolor {Chapter }{DirectoriesPackagePrograms}}{1315}{subsection.76.3.8}%
\contentsline {subsection}{\numberline {76.3.9}\textcolor {Chapter }{CompareVersionNumbers}}{1315}{subsection.76.3.9}%
\contentsline {subsection}{\numberline {76.3.10}\textcolor {Chapter }{DeclareAutoreadableVariables}}{1315}{subsection.76.3.10}%
\contentsline {subsection}{\numberline {76.3.11}\textcolor {Chapter }{Kernel modules in \textsf {GAP} packages}}{1316}{subsection.76.3.11}%
\contentsline {subsection}{\numberline {76.3.12}\textcolor {Chapter }{IsKernelExtensionAvailable}}{1316}{subsection.76.3.12}%
\contentsline {subsection}{\numberline {76.3.13}\textcolor {Chapter }{LoadKernelExtension}}{1316}{subsection.76.3.13}%
\contentsline {subsection}{\numberline {76.3.14}\textcolor {Chapter }{LoadDynamicModule}}{1317}{subsection.76.3.14}%
\contentsline {subsection}{\numberline {76.3.15}\textcolor {Chapter }{The PackageInfo.g File}}{1317}{subsection.76.3.15}%
\contentsline {subsection}{\numberline {76.3.16}\textcolor {Chapter }{ValidatePackageInfo}}{1320}{subsection.76.3.16}%
\contentsline {subsection}{\numberline {76.3.17}\textcolor {Chapter }{ShowPackageVariables}}{1321}{subsection.76.3.17}%
\contentsline {subsection}{\numberline {76.3.18}\textcolor {Chapter }{BibEntry}}{1322}{subsection.76.3.18}%
\contentsline {subsection}{\numberline {76.3.19}\textcolor {Chapter }{Cite}}{1323}{subsection.76.3.19}%
\contentsline {section}{\numberline {76.4}\textcolor {Chapter }{Guidelines for Writing a \textsf {GAP} Package}}{1323}{section.76.4}%
\contentsline {section}{\numberline {76.5}\textcolor {Chapter }{Structure of a \textsf {GAP} Package}}{1323}{section.76.5}%
\contentsline {section}{\numberline {76.6}\textcolor {Chapter }{Writing Documentation and Tools Needed}}{1327}{section.76.6}%
\contentsline {section}{\numberline {76.7}\textcolor {Chapter }{An Example of a \textsf {GAP} Package}}{1328}{section.76.7}%
\contentsline {section}{\numberline {76.8}\textcolor {Chapter }{File Structure}}{1329}{section.76.8}%
\contentsline {section}{\numberline {76.9}\textcolor {Chapter }{Creating the PackageInfo.g File}}{1330}{section.76.9}%
\contentsline {section}{\numberline {76.10}\textcolor {Chapter }{Functions and Variables and Choices of Their Names}}{1330}{section.76.10}%
\contentsline {section}{\numberline {76.11}\textcolor {Chapter }{Package Dependencies (Requesting one \textsf {GAP} Package from within Another)}}{1333}{section.76.11}%
\contentsline {section}{\numberline {76.12}\textcolor {Chapter }{External Dependencies (System packages needed by a \textsf {GAP} Package)}}{1334}{section.76.12}%
\contentsline {section}{\numberline {76.13}\textcolor {Chapter }{Extensions Provided by a Package}}{1335}{section.76.13}%
\contentsline {section}{\numberline {76.14}\textcolor {Chapter }{Declaration and Implementation Part of a Package}}{1336}{section.76.14}%
\contentsline {section}{\numberline {76.15}\textcolor {Chapter }{Autoreadable Variables}}{1337}{section.76.15}%
\contentsline {section}{\numberline {76.16}\textcolor {Chapter }{Standalone Programs in a \textsf {GAP} Package}}{1337}{section.76.16}%
\contentsline {subsection}{\numberline {76.16.1}\textcolor {Chapter }{Installation of \textsf {GAP} Package Binaries}}{1338}{subsection.76.16.1}%
\contentsline {subsection}{\numberline {76.16.2}\textcolor {Chapter }{Test for the Existence of GAP Package Binaries}}{1338}{subsection.76.16.2}%
\contentsline {subsection}{\numberline {76.16.3}\textcolor {Chapter }{Calling of and Communication with External Binaries}}{1339}{subsection.76.16.3}%
\contentsline {section}{\numberline {76.17}\textcolor {Chapter }{Having an InfoClass}}{1340}{section.76.17}%
\contentsline {section}{\numberline {76.18}\textcolor {Chapter }{The Banner}}{1340}{section.76.18}%
\contentsline {section}{\numberline {76.19}\textcolor {Chapter }{Version Numbers}}{1340}{section.76.19}%
\contentsline {section}{\numberline {76.20}\textcolor {Chapter }{Testing a \textsf {GAP} package}}{1341}{section.76.20}%
\contentsline {subsection}{\numberline {76.20.1}\textcolor {Chapter }{Tests files for a GAP package}}{1342}{subsection.76.20.1}%
\contentsline {subsection}{\numberline {76.20.2}\textcolor {Chapter }{Testing \textsf {GAP} package loading}}{1342}{subsection.76.20.2}%
\contentsline {subsection}{\numberline {76.20.3}\textcolor {Chapter }{LoadAllPackages}}{1343}{subsection.76.20.3}%
\contentsline {subsection}{\numberline {76.20.4}\textcolor {Chapter }{Testing a \textsf {GAP} package with the \textsf {GAP} standard test suite}}{1343}{subsection.76.20.4}%
\contentsline {section}{\numberline {76.21}\textcolor {Chapter }{Access to the \textsf {GAP} Development Version}}{1344}{section.76.21}%
\contentsline {section}{\numberline {76.22}\textcolor {Chapter }{Version control and continuous integration for \textsf {GAP} packages}}{1344}{section.76.22}%
\contentsline {section}{\numberline {76.23}\textcolor {Chapter }{Selecting a license for a \textsf {GAP} Package}}{1345}{section.76.23}%
\contentsline {section}{\numberline {76.24}\textcolor {Chapter }{Releasing a GAP Package}}{1345}{section.76.24}%
\contentsline {section}{\numberline {76.25}\textcolor {Chapter }{The homepage of a Package}}{1346}{section.76.25}%
\contentsline {section}{\numberline {76.26}\textcolor {Chapter }{Some things to keep in mind}}{1346}{section.76.26}%
\contentsline {section}{\numberline {76.27}\textcolor {Chapter }{Package release checklists}}{1347}{section.76.27}%
\contentsline {subsection}{\numberline {76.27.1}\textcolor {Chapter }{Checklist for releasing a new package}}{1347}{subsection.76.27.1}%
\contentsline {subsection}{\numberline {76.27.2}\textcolor {Chapter }{Checklist for upgrading the package for the next major release of \textsf {GAP}}}{1348}{subsection.76.27.2}%
\contentsline {chapter}{\numberline {77}\textcolor {Chapter }{Replaced and Removed Command Names}}{1349}{chapter.77}%
\contentsline {section}{\numberline {77.1}\textcolor {Chapter }{Group Actions {\textendash } Name Changes}}{1349}{section.77.1}%
\contentsline {section}{\numberline {77.2}\textcolor {Chapter }{Package Interface {\textendash } Obsolete Functions and Name Changes}}{1350}{section.77.2}%
\contentsline {section}{\numberline {77.3}\textcolor {Chapter }{Normal Forms of Integer Matrices {\textendash } Name Changes}}{1350}{section.77.3}%
\contentsline {section}{\numberline {77.4}\textcolor {Chapter }{Miscellaneous Name Changes or Removed Names}}{1350}{section.77.4}%
\contentsline {subsection}{\numberline {77.4.1}\textcolor {Chapter }{InfoObsolete}}{1351}{subsection.77.4.1}%
\contentsline {section}{\numberline {77.5}\textcolor {Chapter }{The former .gaprc file}}{1352}{section.77.5}%
\contentsline {section}{\numberline {77.6}\textcolor {Chapter }{Semigroup properties}}{1352}{section.77.6}%
\contentsline {chapter}{\numberline {78}\textcolor {Chapter }{Method Selection}}{1353}{chapter.78}%
\contentsline {section}{\numberline {78.1}\textcolor {Chapter }{Operations and Methods}}{1353}{section.78.1}%
\contentsline {subsection}{\numberline {78.1.1}\textcolor {Chapter }{IsOperation}}{1353}{subsection.78.1.1}%
\contentsline {subsection}{\numberline {78.1.2}\textcolor {Chapter }{TypeOfOperation}}{1354}{subsection.78.1.2}%
\contentsline {subsection}{\numberline {78.1.3}\textcolor {Chapter }{ShowDeclarationsOfOperation}}{1354}{subsection.78.1.3}%
\contentsline {subsection}{\numberline {78.1.4}\textcolor {Chapter }{NewOperation}}{1354}{subsection.78.1.4}%
\contentsline {subsection}{\numberline {78.1.5}\textcolor {Chapter }{DeclareOperation}}{1354}{subsection.78.1.5}%
\contentsline {subsection}{\numberline {78.1.6}\textcolor {Chapter }{Tag Based Operations}}{1355}{subsection.78.1.6}%
\contentsline {section}{\numberline {78.2}\textcolor {Chapter }{Constructors}}{1355}{section.78.2}%
\contentsline {subsection}{\numberline {78.2.1}\textcolor {Chapter }{NewConstructor}}{1356}{subsection.78.2.1}%
\contentsline {subsection}{\numberline {78.2.2}\textcolor {Chapter }{DeclareConstructor}}{1358}{subsection.78.2.2}%
\contentsline {section}{\numberline {78.3}\textcolor {Chapter }{Method Installation}}{1358}{section.78.3}%
\contentsline {subsection}{\numberline {78.3.1}\textcolor {Chapter }{InstallMethod}}{1358}{subsection.78.3.1}%
\contentsline {subsection}{\numberline {78.3.2}\textcolor {Chapter }{InstallOtherMethod}}{1359}{subsection.78.3.2}%
\contentsline {subsection}{\numberline {78.3.3}\textcolor {Chapter }{InstallEarlyMethod}}{1359}{subsection.78.3.3}%
\contentsline {subsection}{\numberline {78.3.4}\textcolor {Chapter }{InstallMethodWithRandomSource}}{1359}{subsection.78.3.4}%
\contentsline {section}{\numberline {78.4}\textcolor {Chapter }{Applicable Methods and Method Selection}}{1360}{section.78.4}%
\contentsline {section}{\numberline {78.5}\textcolor {Chapter }{Partial Methods}}{1361}{section.78.5}%
\contentsline {subsection}{\numberline {78.5.1}\textcolor {Chapter }{TryNextMethod}}{1361}{subsection.78.5.1}%
\contentsline {section}{\numberline {78.6}\textcolor {Chapter }{Redispatching}}{1361}{section.78.6}%
\contentsline {subsection}{\numberline {78.6.1}\textcolor {Chapter }{RedispatchOnCondition}}{1362}{subsection.78.6.1}%
\contentsline {section}{\numberline {78.7}\textcolor {Chapter }{Immediate Methods}}{1362}{section.78.7}%
\contentsline {subsection}{\numberline {78.7.1}\textcolor {Chapter }{InstallImmediateMethod}}{1362}{subsection.78.7.1}%
\contentsline {subsection}{\numberline {78.7.2}\textcolor {Chapter }{IsNoImmediateMethodsObject}}{1363}{subsection.78.7.2}%
\contentsline {section}{\numberline {78.8}\textcolor {Chapter }{Logical Implications}}{1363}{section.78.8}%
\contentsline {subsection}{\numberline {78.8.1}\textcolor {Chapter }{InstallTrueMethod}}{1363}{subsection.78.8.1}%
\contentsline {subsection}{\numberline {78.8.2}\textcolor {Chapter }{SuspendMethodReordering}}{1364}{subsection.78.8.2}%
\contentsline {section}{\numberline {78.9}\textcolor {Chapter }{Operations and Mathematical Terms}}{1364}{section.78.9}%
\contentsline {chapter}{\numberline {79}\textcolor {Chapter }{Creating New Objects}}{1366}{chapter.79}%
\contentsline {section}{\numberline {79.1}\textcolor {Chapter }{Creating Objects}}{1366}{section.79.1}%
\contentsline {subsection}{\numberline {79.1.1}\textcolor {Chapter }{Objectify}}{1366}{subsection.79.1.1}%
\contentsline {subsection}{\numberline {79.1.2}\textcolor {Chapter }{ObjectifyWithAttributes}}{1366}{subsection.79.1.2}%
\contentsline {section}{\numberline {79.2}\textcolor {Chapter }{Component Objects}}{1367}{section.79.2}%
\contentsline {subsection}{\numberline {79.2.1}\textcolor {Chapter }{NamesOfComponents}}{1368}{subsection.79.2.1}%
\contentsline {section}{\numberline {79.3}\textcolor {Chapter }{Positional Objects}}{1368}{section.79.3}%
\contentsline {section}{\numberline {79.4}\textcolor {Chapter }{Implementing New List Objects}}{1370}{section.79.4}%
\contentsline {section}{\numberline {79.5}\textcolor {Chapter }{Example {\textendash } Constructing Enumerators}}{1371}{section.79.5}%
\contentsline {section}{\numberline {79.6}\textcolor {Chapter }{Example {\textendash } Constructing Iterators}}{1373}{section.79.6}%
\contentsline {section}{\numberline {79.7}\textcolor {Chapter }{Arithmetic Issues in the Implementation of New Kinds of Lists}}{1374}{section.79.7}%
\contentsline {section}{\numberline {79.8}\textcolor {Chapter }{External Representation}}{1375}{section.79.8}%
\contentsline {subsection}{\numberline {79.8.1}\textcolor {Chapter }{ExtRepOfObj}}{1376}{subsection.79.8.1}%
\contentsline {section}{\numberline {79.9}\textcolor {Chapter }{Mutability and Copying}}{1376}{section.79.9}%
\contentsline {section}{\numberline {79.10}\textcolor {Chapter }{Global Variables in the Library}}{1378}{section.79.10}%
\contentsline {subsection}{\numberline {79.10.1}\textcolor {Chapter }{DeclareGlobalName}}{1379}{subsection.79.10.1}%
\contentsline {subsection}{\numberline {79.10.2}\textcolor {Chapter }{DeclareGlobalVariable}}{1379}{subsection.79.10.2}%
\contentsline {subsection}{\numberline {79.10.3}\textcolor {Chapter }{InstallValue}}{1379}{subsection.79.10.3}%
\contentsline {subsection}{\numberline {79.10.4}\textcolor {Chapter }{FlushCaches}}{1380}{subsection.79.10.4}%
\contentsline {subsection}{\numberline {79.10.5}\textcolor {Chapter }{DeclareGlobalFunction}}{1380}{subsection.79.10.5}%
\contentsline {subsection}{\numberline {79.10.6}\textcolor {Chapter }{DeclareSynonym}}{1381}{subsection.79.10.6}%
\contentsline {section}{\numberline {79.11}\textcolor {Chapter }{Declaration and Implementation Part}}{1381}{section.79.11}%
\contentsline {chapter}{\numberline {80}\textcolor {Chapter }{Examples of Extending the System}}{1383}{chapter.80}%
\contentsline {section}{\numberline {80.1}\textcolor {Chapter }{Addition of a Method}}{1383}{section.80.1}%
\contentsline {section}{\numberline {80.2}\textcolor {Chapter }{Extending the Range of Definition of an Existing Operation}}{1385}{section.80.2}%
\contentsline {section}{\numberline {80.3}\textcolor {Chapter }{Enforcing Property Tests}}{1385}{section.80.3}%
\contentsline {section}{\numberline {80.4}\textcolor {Chapter }{Adding a new Operation}}{1386}{section.80.4}%
\contentsline {section}{\numberline {80.5}\textcolor {Chapter }{Adding a new Attribute}}{1386}{section.80.5}%
\contentsline {section}{\numberline {80.6}\textcolor {Chapter }{Adding a new Representation}}{1387}{section.80.6}%
\contentsline {section}{\numberline {80.7}\textcolor {Chapter }{Components versus Attributes}}{1389}{section.80.7}%
\contentsline {section}{\numberline {80.8}\textcolor {Chapter }{Adding new Concepts}}{1389}{section.80.8}%
\contentsline {subsection}{\numberline {80.8.1}\textcolor {Chapter }{Example: M\texttt {\symbol {45}}groups}}{1390}{subsection.80.8.1}%
\contentsline {subsection}{\numberline {80.8.2}\textcolor {Chapter }{Example: Groups with a word length}}{1391}{subsection.80.8.2}%
\contentsline {subsection}{\numberline {80.8.3}\textcolor {Chapter }{Example: Groups with a decomposition as semidirect product}}{1391}{subsection.80.8.3}%
\contentsline {section}{\numberline {80.9}\textcolor {Chapter }{Creating Own Arithmetic Objects}}{1392}{section.80.9}%
\contentsline {subsection}{\numberline {80.9.1}\textcolor {Chapter }{ArithmeticElementCreator}}{1392}{subsection.80.9.1}%
\contentsline {subsection}{\numberline {80.9.2}\textcolor {Chapter }{Example: ArithmeticElementCreator}}{1393}{subsection.80.9.2}%
\contentsline {chapter}{\numberline {81}\textcolor {Chapter }{An Example {\textendash } Residue Class Rings}}{1395}{chapter.81}%
\contentsline {section}{\numberline {81.1}\textcolor {Chapter }{A First Attempt to Implement Elements of Residue Class Rings}}{1395}{section.81.1}%
\contentsline {section}{\numberline {81.2}\textcolor {Chapter }{Why Proceed in a Different Way?}}{1396}{section.81.2}%
\contentsline {section}{\numberline {81.3}\textcolor {Chapter }{A Second Attempt to Implement Elements of Residue Class Rings}}{1397}{section.81.3}%
\contentsline {section}{\numberline {81.4}\textcolor {Chapter }{Compatibility of Residue Class Rings with Prime Fields}}{1408}{section.81.4}%
\contentsline {section}{\numberline {81.5}\textcolor {Chapter }{Further Improvements in Implementing Residue Class Rings}}{1414}{section.81.5}%
\contentsline {chapter}{\numberline {82}\textcolor {Chapter }{An Example {\textendash } Designing Arithmetic Operations}}{1416}{chapter.82}%
\contentsline {section}{\numberline {82.1}\textcolor {Chapter }{New Arithmetic Operations vs. New Objects}}{1416}{section.82.1}%
\contentsline {section}{\numberline {82.2}\textcolor {Chapter }{Designing new Multiplicative Objects}}{1417}{section.82.2}%
\contentsline {chapter}{\numberline {83}\textcolor {Chapter }{Library Files}}{1424}{chapter.83}%
\contentsline {section}{\numberline {83.1}\textcolor {Chapter }{File Types}}{1424}{section.83.1}%
\contentsline {section}{\numberline {83.2}\textcolor {Chapter }{Finding Implementations in the Library}}{1424}{section.83.2}%
\contentsline {section}{\numberline {83.3}\textcolor {Chapter }{Undocumented Variables}}{1425}{section.83.3}%
\contentsline {chapter}{\numberline {84}\textcolor {Chapter }{Interface to the GAP Help System}}{1427}{chapter.84}%
\contentsline {section}{\numberline {84.1}\textcolor {Chapter }{Installing and Removing a Help Book}}{1427}{section.84.1}%
\contentsline {subsection}{\numberline {84.1.1}\textcolor {Chapter }{HELP{\textunderscore }ADD{\textunderscore }BOOK}}{1427}{subsection.84.1.1}%
\contentsline {subsection}{\numberline {84.1.2}\textcolor {Chapter }{HELP{\textunderscore }REMOVE{\textunderscore }BOOK}}{1428}{subsection.84.1.2}%
\contentsline {section}{\numberline {84.2}\textcolor {Chapter }{The manual.six File}}{1428}{section.84.2}%
\contentsline {section}{\numberline {84.3}\textcolor {Chapter }{The Help Book Handler}}{1428}{section.84.3}%
\contentsline {section}{\numberline {84.4}\textcolor {Chapter }{Introducing new Viewer for the Online Help}}{1430}{section.84.4}%
\contentsline {subsection}{\numberline {84.4.1}\textcolor {Chapter }{HELP{\textunderscore }VIEWER{\textunderscore }INFO}}{1430}{subsection.84.4.1}%
\contentsline {chapter}{\numberline {85}\textcolor {Chapter }{Function\texttt {\symbol {45}}Operation\texttt {\symbol {45}}Attribute Triples}}{1431}{chapter.85}%
\contentsline {section}{\numberline {85.1}\textcolor {Chapter }{Key Dependent Operations}}{1431}{section.85.1}%
\contentsline {subsection}{\numberline {85.1.1}\textcolor {Chapter }{KeyDependentOperation}}{1431}{subsection.85.1.1}%
\contentsline {section}{\numberline {85.2}\textcolor {Chapter }{In Parent Attributes}}{1433}{section.85.2}%
\contentsline {subsection}{\numberline {85.2.1}\textcolor {Chapter }{InParentFOA}}{1433}{subsection.85.2.1}%
\contentsline {section}{\numberline {85.3}\textcolor {Chapter }{Operation Functions}}{1433}{section.85.3}%
\contentsline {subsection}{\numberline {85.3.1}\textcolor {Chapter }{OrbitsishOperation}}{1434}{subsection.85.3.1}%
\contentsline {subsection}{\numberline {85.3.2}\textcolor {Chapter }{OrbitishFO}}{1435}{subsection.85.3.2}%
\contentsline {subsection}{\numberline {85.3.3}\textcolor {Chapter }{Example: Orbit and OrbitOp}}{1435}{subsection.85.3.3}%
\contentsline {chapter}{\numberline {86}\textcolor {Chapter }{Weak Pointers}}{1437}{chapter.86}%
\contentsline {section}{\numberline {86.1}\textcolor {Chapter }{Weak Pointer Objects}}{1437}{section.86.1}%
\contentsline {subsection}{\numberline {86.1.1}\textcolor {Chapter }{WeakPointerObj}}{1437}{subsection.86.1.1}%
\contentsline {section}{\numberline {86.2}\textcolor {Chapter }{Low Level Access Functions for Weak Pointer Objects}}{1438}{section.86.2}%
\contentsline {subsection}{\numberline {86.2.1}\textcolor {Chapter }{SetElmWPObj}}{1438}{subsection.86.2.1}%
\contentsline {section}{\numberline {86.3}\textcolor {Chapter }{Accessing Weak Pointer Objects as Lists}}{1439}{section.86.3}%
\contentsline {section}{\numberline {86.4}\textcolor {Chapter }{Copying Weak Pointer Objects}}{1439}{section.86.4}%
\contentsline {chapter}{\numberline {87}\textcolor {Chapter }{More about Stabilizer Chains}}{1441}{chapter.87}%
\contentsline {section}{\numberline {87.1}\textcolor {Chapter }{Generalized Conjugation Technique}}{1441}{section.87.1}%
\contentsline {section}{\numberline {87.2}\textcolor {Chapter }{The General Backtrack Algorithm with Ordered Partitions}}{1442}{section.87.2}%
\contentsline {subsection}{\numberline {87.2.1}\textcolor {Chapter }{Internal representation of ordered partitions}}{1442}{subsection.87.2.1}%
\contentsline {subsection}{\numberline {87.2.2}\textcolor {Chapter }{Functions for setting up an R\texttt {\symbol {45}}base}}{1443}{subsection.87.2.2}%
\contentsline {subsection}{\numberline {87.2.3}\textcolor {Chapter }{Refinement functions for the backtrack search}}{1446}{subsection.87.2.3}%
\contentsline {subsection}{\numberline {87.2.4}\textcolor {Chapter }{Functions for meeting ordered partitions}}{1447}{subsection.87.2.4}%
\contentsline {section}{\numberline {87.3}\textcolor {Chapter }{Stabilizer Chains for Automorphisms Acting on Enumerators}}{1448}{section.87.3}%
\contentsline {subsection}{\numberline {87.3.1}\textcolor {Chapter }{An operation domain for automorphisms}}{1449}{subsection.87.3.1}%
\contentsline {subsection}{\numberline {87.3.2}\textcolor {Chapter }{Enumerators for cosets of characteristic factors}}{1450}{subsection.87.3.2}%
\contentsline {subsection}{\numberline {87.3.3}\textcolor {Chapter }{Making automorphisms act on such enumerators}}{1451}{subsection.87.3.3}%
\contentsline {chapter}{References}{1461}{chapter*.3}%
\contentsline {chapter}{Index}{1462}{section*.4}%
