\contentsline {chapter}{\numberline {1}About GAP}{57}
\contentsline {section}{\numberline {1.1}About Conventions}{58}
\contentsline {section}{\numberline {1.2}About Starting and Leaving GAP}{58}
\contentsline {section}{\numberline {1.3}About First Steps}{59}
\contentsline {section}{\numberline {1.4}About Help}{60}
\contentsline {section}{\numberline {1.5}About Syntax Errors}{60}
\contentsline {section}{\numberline {1.6}About Constants and Operators}{60}
\contentsline {section}{\numberline {1.7}About Variables and Assignments}{62}
\contentsline {section}{\numberline {1.8}About Functions}{64}
\contentsline {section}{\numberline {1.9}About Lists}{65}
\contentsline {section}{\numberline {1.10}About Identical Lists}{67}
\contentsline {section}{\numberline {1.11}About Sets}{69}
\contentsline {section}{\numberline {1.12}About Vectors and Matrices}{70}
\contentsline {section}{\numberline {1.13}About Records}{72}
\contentsline {section}{\numberline {1.14}About Ranges}{73}
\contentsline {section}{\numberline {1.15}About Loops}{74}
\contentsline {section}{\numberline {1.16}About Further List Operations}{76}
\contentsline {section}{\numberline {1.17}About Writing Functions}{77}
\contentsline {section}{\numberline {1.18}About Groups}{81}
\contentsline {section}{\numberline {1.19}About Operations of Groups}{89}
\contentsline {section}{\numberline {1.20}About Finitely Presented Groups and Presentations}{97}
\contentsline {section}{\numberline {1.21}About Fields}{102}
\contentsline {section}{\numberline {1.22}About Matrix Groups}{106}
\contentsline {section}{\numberline {1.23}About Domains and Categories}{107}
\contentsline {section}{\numberline {1.24}About Mappings and Homomorphisms}{115}
\contentsline {section}{\numberline {1.25}About Character Tables}{120}
\contentsline {section}{\numberline {1.26}About Group Libraries}{135}
\contentsline {section}{\numberline {1.27}About the Implementation of Domains}{141}
\contentsline {section}{\numberline {1.28}About Defining New Domains}{150}
\contentsline {section}{\numberline {1.29}About Defining New Parametrized Domains}{158}
\contentsline {section}{\numberline {1.30}About Defining New Group Elements}{162}
\contentsline {chapter}{\numberline {2}The Programming Language}{179}
\contentsline {section}{\numberline {2.1}Lexical Structure}{180}
\contentsline {section}{\numberline {2.2}Symbols}{180}
\contentsline {section}{\numberline {2.3}Whitespaces}{181}
\contentsline {section}{\numberline {2.4}Keywords}{182}
\contentsline {section}{\numberline {2.5}Identifiers}{182}
\contentsline {section}{\numberline {2.6}Expressions}{182}
\contentsline {section}{\numberline {2.7}Variables}{183}
\contentsline {section}{\numberline {2.8}Function Calls}{184}
\contentsline {section}{\numberline {2.9}Comparisons}{185}
\contentsline {section}{\numberline {2.10}Operations}{186}
\contentsline {section}{\numberline {2.11}Statements}{186}
\contentsline {section}{\numberline {2.12}Assignments}{187}
\contentsline {section}{\numberline {2.13}Procedure Calls}{188}
\contentsline {section}{\numberline {2.14}If}{188}
\contentsline {section}{\numberline {2.15}While}{189}
\contentsline {section}{\numberline {2.16}Repeat}{189}
\contentsline {section}{\numberline {2.17}For}{190}
\contentsline {section}{\numberline {2.18}Functions}{191}
\contentsline {section}{\numberline {2.19}Return}{193}
\contentsline {section}{\numberline {2.20}The Syntax in BNF}{193}
\contentsline {chapter}{\numberline {3}Environment}{197}
\contentsline {section}{\numberline {3.1}Main Loop}{197}
\contentsline {section}{\numberline {3.2}Break Loops}{199}
\contentsline {section}{\numberline {3.3}Error}{199}
\contentsline {section}{\numberline {3.4}Line Editing}{199}
\contentsline {section}{\numberline {3.5}Help}{201}
\contentsline {section}{\numberline {3.6}Reading Sections}{201}
\contentsline {section}{\numberline {3.7}Format of Sections}{201}
\contentsline {section}{\numberline {3.8}Browsing through the Sections}{202}
\contentsline {section}{\numberline {3.9}Redisplaying a Section}{203}
\contentsline {section}{\numberline {3.10}Abbreviating Section Names}{203}
\contentsline {section}{\numberline {3.11}Help Index}{203}
\contentsline {section}{\numberline {3.12}Read}{204}
\contentsline {section}{\numberline {3.13}ReadLib}{205}
\contentsline {section}{\numberline {3.14}Print}{205}
\contentsline {section}{\numberline {3.15}PrintTo}{205}
\contentsline {section}{\numberline {3.16}AppendTo}{206}
\contentsline {section}{\numberline {3.17}LogTo}{206}
\contentsline {section}{\numberline {3.18}LogInputTo}{206}
\contentsline {section}{\numberline {3.19}SizeScreen}{206}
\contentsline {section}{\numberline {3.20}Runtime}{206}
\contentsline {section}{\numberline {3.21}Profile}{207}
\contentsline {section}{\numberline {3.22}Exec}{208}
\contentsline {section}{\numberline {3.23}Edit}{208}
\contentsline {chapter}{\numberline {4}Domains}{209}
\contentsline {section}{\numberline {4.1}Domain Records}{210}
\contentsline {section}{\numberline {4.2}Dispatchers}{210}
\contentsline {section}{\numberline {4.3}More about Dispatchers}{211}
\contentsline {section}{\numberline {4.4}An Example of a Computation in a Domain}{212}
\contentsline {section}{\numberline {4.5}Domain}{213}
\contentsline {section}{\numberline {4.6}Elements}{214}
\contentsline {section}{\numberline {4.7}Comparisons of Domains}{214}
\contentsline {section}{\numberline {4.8}Membership Test for Domains}{216}
\contentsline {section}{\numberline {4.9}IsFinite}{216}
\contentsline {section}{\numberline {4.10}Size}{217}
\contentsline {section}{\numberline {4.11}IsSubset}{217}
\contentsline {section}{\numberline {4.12}Intersection}{217}
\contentsline {section}{\numberline {4.13}Union}{218}
\contentsline {section}{\numberline {4.14}Difference}{219}
\contentsline {section}{\numberline {4.15}Representative}{220}
\contentsline {section}{\numberline {4.16}Random}{220}
\contentsline {chapter}{\numberline {5}Rings}{223}
\contentsline {section}{\numberline {5.1}IsRing}{223}
\contentsline {section}{\numberline {5.2}Ring}{224}
\contentsline {section}{\numberline {5.3}DefaultRing}{224}
\contentsline {section}{\numberline {5.4}Comparisons of Ring Elements}{225}
\contentsline {section}{\numberline {5.5}Operations for Ring Elements}{225}
\contentsline {section}{\numberline {5.6}Quotient}{226}
\contentsline {section}{\numberline {5.7}IsCommutativeRing}{226}
\contentsline {section}{\numberline {5.8}IsIntegralRing}{226}
\contentsline {section}{\numberline {5.9}IsUniqueFactorizationRing}{227}
\contentsline {section}{\numberline {5.10}IsEuclideanRing}{227}
\contentsline {section}{\numberline {5.11}IsUnit}{228}
\contentsline {section}{\numberline {5.12}Units}{228}
\contentsline {section}{\numberline {5.13}IsAssociated}{228}
\contentsline {section}{\numberline {5.14}StandardAssociate}{229}
\contentsline {section}{\numberline {5.15}Associates}{229}
\contentsline {section}{\numberline {5.16}IsIrreducible}{230}
\contentsline {section}{\numberline {5.17}IsPrime}{230}
\contentsline {section}{\numberline {5.18}Factors}{230}
\contentsline {section}{\numberline {5.19}EuclideanDegree}{231}
\contentsline {section}{\numberline {5.20}EuclideanRemainder}{231}
\contentsline {section}{\numberline {5.21}EuclideanQuotient}{232}
\contentsline {section}{\numberline {5.22}QuotientRemainder}{232}
\contentsline {section}{\numberline {5.23}Mod}{233}
\contentsline {section}{\numberline {5.24}QuotientMod}{233}
\contentsline {section}{\numberline {5.25}PowerMod}{234}
\contentsline {section}{\numberline {5.26}Gcd}{234}
\contentsline {section}{\numberline {5.27}GcdRepresentation}{235}
\contentsline {section}{\numberline {5.28}Lcm}{235}
\contentsline {section}{\numberline {5.29}Ring Records}{236}
\contentsline {chapter}{\numberline {6}Fields}{239}
\contentsline {section}{\numberline {6.1}IsField}{239}
\contentsline {section}{\numberline {6.2}Field}{240}
\contentsline {section}{\numberline {6.3}DefaultField}{240}
\contentsline {section}{\numberline {6.4}Fields over Subfields}{241}
\contentsline {section}{\numberline {6.5}Comparisons of Field Elements}{241}
\contentsline {section}{\numberline {6.6}Operations for Field Elements}{242}
\contentsline {section}{\numberline {6.7}GaloisGroup}{242}
\contentsline {section}{\numberline {6.8}MinPol}{243}
\contentsline {section}{\numberline {6.9}CharPol}{243}
\contentsline {section}{\numberline {6.10}Norm}{244}
\contentsline {section}{\numberline {6.11}Trace}{245}
\contentsline {section}{\numberline {6.12}Conjugates}{245}
\contentsline {section}{\numberline {6.13}Field Homomorphisms}{246}
\contentsline {section}{\numberline {6.14}IsFieldHomomorphism}{246}
\contentsline {section}{\numberline {6.15}KernelFieldHomomorphism}{247}
\contentsline {section}{\numberline {6.16}Mapping Functions for Field Homomorphisms}{247}
\contentsline {section}{\numberline {6.17}Field Records}{248}
\contentsline {chapter}{\numberline {7}Groups}{249}
\contentsline {section}{\numberline {7.1}Group Elements}{250}
\contentsline {section}{\numberline {7.2}Comparisons of Group Elements}{250}
\contentsline {section}{\numberline {7.3}Operations for Group Elements}{250}
\contentsline {section}{\numberline {7.4}IsGroupElement}{251}
\contentsline {section}{\numberline {7.5}Order}{252}
\contentsline {section}{\numberline {7.6}More about Groups and Subgroups}{252}
\contentsline {section}{\numberline {7.7}IsParent}{253}
\contentsline {section}{\numberline {7.8}Parent}{253}
\contentsline {section}{\numberline {7.9}Group}{254}
\contentsline {section}{\numberline {7.10}AsGroup}{255}
\contentsline {section}{\numberline {7.11}IsGroup}{255}
\contentsline {section}{\numberline {7.12}Subgroup}{255}
\contentsline {section}{\numberline {7.13}AsSubgroup}{256}
\contentsline {section}{\numberline {7.14}Subgroups}{256}
\contentsline {section}{\numberline {7.15}Agemo}{256}
\contentsline {section}{\numberline {7.16}Centralizer}{257}
\contentsline {section}{\numberline {7.17}Centre}{257}
\contentsline {section}{\numberline {7.18}Closure}{258}
\contentsline {section}{\numberline {7.19}CommutatorSubgroup}{259}
\contentsline {section}{\numberline {7.20}ConjugateSubgroup}{259}
\contentsline {section}{\numberline {7.21}Core}{259}
\contentsline {section}{\numberline {7.22}DerivedSubgroup}{260}
\contentsline {section}{\numberline {7.23}FittingSubgroup}{260}
\contentsline {section}{\numberline {7.24}FrattiniSubgroup}{261}
\contentsline {section}{\numberline {7.25}NormalClosure}{261}
\contentsline {section}{\numberline {7.26}NormalIntersection}{261}
\contentsline {section}{\numberline {7.27}Normalizer}{261}
\contentsline {section}{\numberline {7.28}PCore}{262}
\contentsline {section}{\numberline {7.29}PrefrattiniSubgroup}{262}
\contentsline {section}{\numberline {7.30}Radical}{263}
\contentsline {section}{\numberline {7.31}SylowSubgroup}{263}
\contentsline {section}{\numberline {7.32}TrivialSubgroup}{263}
\contentsline {section}{\numberline {7.33}FactorGroup}{263}
\contentsline {section}{\numberline {7.34}FactorGroupElement}{264}
\contentsline {section}{\numberline {7.35}CommutatorFactorGroup}{265}
\contentsline {section}{\numberline {7.36}Series of Subgroups}{265}
\contentsline {section}{\numberline {7.37}DerivedSeries}{265}
\contentsline {section}{\numberline {7.38}CompositionSeries}{266}
\contentsline {section}{\numberline {7.39}ElementaryAbelianSeries}{266}
\contentsline {section}{\numberline {7.40}JenningsSeries}{266}
\contentsline {section}{\numberline {7.41}LowerCentralSeries}{267}
\contentsline {section}{\numberline {7.42}PCentralSeries}{267}
\contentsline {section}{\numberline {7.43}SubnormalSeries}{267}
\contentsline {section}{\numberline {7.44}UpperCentralSeries}{268}
\contentsline {section}{\numberline {7.45}Properties and Property Tests}{268}
\contentsline {section}{\numberline {7.46}AbelianInvariants}{269}
\contentsline {section}{\numberline {7.47}DimensionsLoewyFactors}{269}
\contentsline {section}{\numberline {7.48}EulerianFunction}{270}
\contentsline {section}{\numberline {7.49}Exponent}{270}
\contentsline {section}{\numberline {7.50}Factorization}{270}
\contentsline {section}{\numberline {7.51}Index}{271}
\contentsline {section}{\numberline {7.52}IsAbelian}{271}
\contentsline {section}{\numberline {7.53}IsCentral}{271}
\contentsline {section}{\numberline {7.54}IsConjugate}{272}
\contentsline {section}{\numberline {7.55}IsCyclic}{272}
\contentsline {section}{\numberline {7.56}IsElementaryAbelian}{272}
\contentsline {section}{\numberline {7.57}IsNilpotent}{273}
\contentsline {section}{\numberline {7.58}IsNormal}{273}
\contentsline {section}{\numberline {7.59}IsPerfect}{274}
\contentsline {section}{\numberline {7.60}IsSimple}{274}
\contentsline {section}{\numberline {7.61}IsSolvable}{274}
\contentsline {section}{\numberline {7.62}IsSubgroup}{275}
\contentsline {section}{\numberline {7.63}IsSubnormal}{275}
\contentsline {section}{\numberline {7.64}IsTrivial for Groups}{276}
\contentsline {section}{\numberline {7.65}GroupId}{276}
\contentsline {section}{\numberline {7.66}PermutationCharacter}{279}
\contentsline {section}{\numberline {7.67}Conjugacy Classes}{279}
\contentsline {section}{\numberline {7.68}ConjugacyClasses}{279}
\contentsline {section}{\numberline {7.69}ConjugacyClass}{280}
\contentsline {section}{\numberline {7.70}IsConjugacyClass}{280}
\contentsline {section}{\numberline {7.71}Set Functions for Conjugacy Classes}{281}
\contentsline {section}{\numberline {7.72}Conjugacy Class Records}{281}
\contentsline {section}{\numberline {7.73}ConjugacyClassesSubgroups}{282}
\contentsline {section}{\numberline {7.74}Lattice}{283}
\contentsline {section}{\numberline {7.75}ConjugacyClassSubgroups}{288}
\contentsline {section}{\numberline {7.76}IsConjugacyClassSubgroups}{289}
\contentsline {section}{\numberline {7.77}Set Functions for Subgroup Conjugacy Classes}{289}
\contentsline {section}{\numberline {7.78}Subgroup Conjugacy Class Records}{290}
\contentsline {section}{\numberline {7.79}ConjugacyClassesMaximalSubgroups}{290}
\contentsline {section}{\numberline {7.80}MaximalSubgroups}{291}
\contentsline {section}{\numberline {7.81}NormalSubgroups}{291}
\contentsline {section}{\numberline {7.82}ConjugateSubgroups}{292}
\contentsline {section}{\numberline {7.83}Cosets of Subgroups}{292}
\contentsline {section}{\numberline {7.84}RightCosets}{292}
\contentsline {section}{\numberline {7.85}RightCoset}{293}
\contentsline {section}{\numberline {7.86}IsRightCoset}{293}
\contentsline {section}{\numberline {7.87}Set Functions for Right Cosets}{294}
\contentsline {section}{\numberline {7.88}Right Cosets Records}{295}
\contentsline {section}{\numberline {7.89}LeftCosets}{295}
\contentsline {section}{\numberline {7.90}LeftCoset}{296}
\contentsline {section}{\numberline {7.91}IsLeftCoset}{296}
\contentsline {section}{\numberline {7.92}DoubleCosets}{297}
\contentsline {section}{\numberline {7.93}DoubleCoset}{297}
\contentsline {section}{\numberline {7.94}IsDoubleCoset}{298}
\contentsline {section}{\numberline {7.95}Set Functions for Double Cosets}{298}
\contentsline {section}{\numberline {7.96}Double Coset Records}{299}
\contentsline {section}{\numberline {7.97}Group Constructions}{300}
\contentsline {section}{\numberline {7.98}DirectProduct}{300}
\contentsline {section}{\numberline {7.99}DirectProduct for Groups}{301}
\contentsline {section}{\numberline {7.100}SemidirectProduct}{301}
\contentsline {section}{\numberline {7.101}SemidirectProduct for Groups}{302}
\contentsline {section}{\numberline {7.102}SubdirectProduct}{302}
\contentsline {section}{\numberline {7.103}WreathProduct}{303}
\contentsline {section}{\numberline {7.104}WreathProduct for Groups}{304}
\contentsline {section}{\numberline {7.105}Group Homomorphisms}{304}
\contentsline {section}{\numberline {7.106}IsGroupHomomorphism}{305}
\contentsline {section}{\numberline {7.107}KernelGroupHomomorphism}{305}
\contentsline {section}{\numberline {7.108}Mapping Functions for Group Homomorphisms}{306}
\contentsline {section}{\numberline {7.109}NaturalHomomorphism}{307}
\contentsline {section}{\numberline {7.110}ConjugationGroupHomomorphism}{308}
\contentsline {section}{\numberline {7.111}InnerAutomorphism}{309}
\contentsline {section}{\numberline {7.112}GroupHomomorphismByImages}{309}
\contentsline {section}{\numberline {7.113}Set Functions for Groups}{311}
\contentsline {section}{\numberline {7.114}Elements for Groups}{311}
\contentsline {section}{\numberline {7.115}Intersection for Groups}{312}
\contentsline {section}{\numberline {7.116}Operations for Groups}{312}
\contentsline {section}{\numberline {7.117}Group Records}{313}
\contentsline {chapter}{\numberline {8}Operations of Groups}{317}
\contentsline {section}{\numberline {8.1}Other Operations}{318}
\contentsline {section}{\numberline {8.2}Cycle}{319}
\contentsline {section}{\numberline {8.3}CycleLength}{319}
\contentsline {section}{\numberline {8.4}Cycles}{320}
\contentsline {section}{\numberline {8.5}CycleLengths}{320}
\contentsline {section}{\numberline {8.6}Permutation}{321}
\contentsline {section}{\numberline {8.7}IsFixpoint}{322}
\contentsline {section}{\numberline {8.8}IsFixpointFree}{322}
\contentsline {section}{\numberline {8.9}DegreeOperation}{323}
\contentsline {section}{\numberline {8.10}IsTransitive}{323}
\contentsline {section}{\numberline {8.11}Transitivity}{324}
\contentsline {section}{\numberline {8.12}IsRegular}{325}
\contentsline {section}{\numberline {8.13}IsSemiRegular}{326}
\contentsline {section}{\numberline {8.14}Orbit}{326}
\contentsline {section}{\numberline {8.15}OrbitLength}{327}
\contentsline {section}{\numberline {8.16}Orbits}{328}
\contentsline {section}{\numberline {8.17}OrbitLengths}{329}
\contentsline {section}{\numberline {8.18}Operation}{329}
\contentsline {section}{\numberline {8.19}OperationHomomorphism}{330}
\contentsline {section}{\numberline {8.20}Blocks}{331}
\contentsline {section}{\numberline {8.21}IsPrimitive}{332}
\contentsline {section}{\numberline {8.22}Stabilizer}{332}
\contentsline {section}{\numberline {8.23}RepresentativeOperation}{333}
\contentsline {section}{\numberline {8.24}RepresentativesOperation}{334}
\contentsline {section}{\numberline {8.25}IsEquivalentOperation}{335}
\contentsline {chapter}{\numberline {9}Vector Spaces}{337}
\contentsline {section}{\numberline {9.1}VectorSpace}{337}
\contentsline {section}{\numberline {9.2}IsVectorSpace}{338}
\contentsline {section}{\numberline {9.3}Vector Space Records}{338}
\contentsline {section}{\numberline {9.4}Set Functions for Vector Spaces}{339}
\contentsline {section}{\numberline {9.5}IsSubspace}{339}
\contentsline {section}{\numberline {9.6}Base}{339}
\contentsline {section}{\numberline {9.7}AddBase}{340}
\contentsline {section}{\numberline {9.8}Dimension}{341}
\contentsline {section}{\numberline {9.9}LinearCombination}{341}
\contentsline {section}{\numberline {9.10}Coefficients}{342}
\contentsline {chapter}{\numberline {10}Integers}{343}
\contentsline {section}{\numberline {10.1}Comparisons of Integers}{344}
\contentsline {section}{\numberline {10.2}Operations for Integers}{344}
\contentsline {section}{\numberline {10.3}QuoInt}{345}
\contentsline {section}{\numberline {10.4}RemInt}{345}
\contentsline {section}{\numberline {10.5}IsInt}{346}
\contentsline {section}{\numberline {10.6}Int}{346}
\contentsline {section}{\numberline {10.7}AbsInt}{346}
\contentsline {section}{\numberline {10.8}SignInt}{346}
\contentsline {section}{\numberline {10.9}ChineseRem}{347}
\contentsline {section}{\numberline {10.10}LogInt}{347}
\contentsline {section}{\numberline {10.11}RootInt}{347}
\contentsline {section}{\numberline {10.12}SmallestRootInt}{348}
\contentsline {section}{\numberline {10.13}Set Functions for Integers}{348}
\contentsline {section}{\numberline {10.14}Ring Functions for Integers}{349}
\contentsline {section}{\numberline {10.15}Primes}{350}
\contentsline {section}{\numberline {10.16}IsPrimeInt}{350}
\contentsline {section}{\numberline {10.17}IsPrimePowerInt}{351}
\contentsline {section}{\numberline {10.18}NextPrimeInt}{351}
\contentsline {section}{\numberline {10.19}PrevPrimeInt}{352}
\contentsline {section}{\numberline {10.20}FactorsInt}{352}
\contentsline {section}{\numberline {10.21}DivisorsInt}{352}
\contentsline {section}{\numberline {10.22}Sigma}{353}
\contentsline {section}{\numberline {10.23}Tau}{353}
\contentsline {section}{\numberline {10.24}MoebiusMu}{354}
\contentsline {chapter}{\numberline {11}Number Theory}{355}
\contentsline {section}{\numberline {11.1}PrimeResidues}{355}
\contentsline {section}{\numberline {11.2}Phi}{356}
\contentsline {section}{\numberline {11.3}Lambda}{356}
\contentsline {section}{\numberline {11.4}OrderMod}{357}
\contentsline {section}{\numberline {11.5}IsPrimitiveRootMod}{357}
\contentsline {section}{\numberline {11.6}PrimitiveRootMod}{358}
\contentsline {section}{\numberline {11.7}Jacobi}{358}
\contentsline {section}{\numberline {11.8}Legendre}{358}
\contentsline {section}{\numberline {11.9}RootMod}{359}
\contentsline {section}{\numberline {11.10}RootsUnityMod}{359}
\contentsline {chapter}{\numberline {12}Rationals}{361}
\contentsline {section}{\numberline {12.1}IsRat}{361}
\contentsline {section}{\numberline {12.2}Numerator}{362}
\contentsline {section}{\numberline {12.3}Denominator}{362}
\contentsline {section}{\numberline {12.4}Comparisons of Rationals}{363}
\contentsline {section}{\numberline {12.5}Operations for Rationals}{363}
\contentsline {section}{\numberline {12.6}Set Functions for Rationals}{364}
\contentsline {section}{\numberline {12.7}Field Functions for Rationals}{364}
\contentsline {chapter}{\numberline {13}Cyclotomics}{365}
\contentsline {section}{\numberline {13.1}More about Cyclotomics}{365}
\contentsline {section}{\numberline {13.2}Cyclotomic Integers}{366}
\contentsline {section}{\numberline {13.3}IntCyc}{367}
\contentsline {section}{\numberline {13.4}RoundCyc}{367}
\contentsline {section}{\numberline {13.5}IsCyc}{367}
\contentsline {section}{\numberline {13.6}IsCycInt}{367}
\contentsline {section}{\numberline {13.7}NofCyc}{368}
\contentsline {section}{\numberline {13.8}CoeffsCyc}{368}
\contentsline {section}{\numberline {13.9}Comparisons of Cyclotomics}{368}
\contentsline {section}{\numberline {13.10}Operations for Cyclotomics}{369}
\contentsline {section}{\numberline {13.11}GaloisCyc}{369}
\contentsline {section}{\numberline {13.12}ATLAS irrationalities}{370}
\contentsline {section}{\numberline {13.13}StarCyc}{371}
\contentsline {section}{\numberline {13.14}Quadratic}{371}
\contentsline {section}{\numberline {13.15}GaloisMat}{372}
\contentsline {section}{\numberline {13.16}RationalizedMat}{373}
\contentsline {chapter}{\numberline {14}Gaussians}{375}
\contentsline {section}{\numberline {14.1}Comparisons of Gaussians}{375}
\contentsline {section}{\numberline {14.2}Operations for Gaussians}{376}
\contentsline {section}{\numberline {14.3}IsGaussRat}{377}
\contentsline {section}{\numberline {14.4}IsGaussInt}{377}
\contentsline {section}{\numberline {14.5}Set Functions for Gaussians}{377}
\contentsline {section}{\numberline {14.6}Field Functions for Gaussian Rationals}{378}
\contentsline {section}{\numberline {14.7}Ring Functions for Gaussian Integers}{378}
\contentsline {section}{\numberline {14.8}TwoSquares}{379}
\contentsline {chapter}{\numberline {15}Subfields of Cyclotomic Fields}{381}
\contentsline {section}{\numberline {15.1}IsNumberField}{382}
\contentsline {section}{\numberline {15.2}IsCyclotomicField}{382}
\contentsline {section}{\numberline {15.3}Number Field Records}{382}
\contentsline {section}{\numberline {15.4}Cyclotomic Field Records}{383}
\contentsline {section}{\numberline {15.5}DefaultField and Field for Cyclotomics}{384}
\contentsline {section}{\numberline {15.6}DefaultRing and Ring for Cyclotomic Integers}{385}
\contentsline {section}{\numberline {15.7}GeneratorsPrimeResidues}{385}
\contentsline {section}{\numberline {15.8}GaloisGroup for Number Fields}{386}
\contentsline {section}{\numberline {15.9}ZumbroichBase}{386}
\contentsline {section}{\numberline {15.10}Integral Bases for Number Fields}{387}
\contentsline {section}{\numberline {15.11}NormalBaseNumberField}{388}
\contentsline {section}{\numberline {15.12}Coefficients for Number Fields}{388}
\contentsline {section}{\numberline {15.13}Domain Functions for Number Fields}{389}
\contentsline {chapter}{\numberline {16}Algebraic extensions of fields}{391}
\contentsline {section}{\numberline {16.1}AlgebraicExtension}{391}
\contentsline {section}{\numberline {16.2}IsAlgebraicExtension}{392}
\contentsline {section}{\numberline {16.3}RootOf}{392}
\contentsline {section}{\numberline {16.4}Algebraic Extension Elements}{392}
\contentsline {section}{\numberline {16.5}Set functions for Algebraic Extensions}{392}
\contentsline {section}{\numberline {16.6}IsNormalExtension}{393}
\contentsline {section}{\numberline {16.7}MinpolFactors}{393}
\contentsline {section}{\numberline {16.8}GaloisGroup for Extension Fields}{393}
\contentsline {section}{\numberline {16.9}ExtensionAutomorphism}{394}
\contentsline {section}{\numberline {16.10}Field functions for Algebraic Extensions}{394}
\contentsline {section}{\numberline {16.11}Algebraic Extension Records}{395}
\contentsline {section}{\numberline {16.12}Extension Element Records}{395}
\contentsline {section}{\numberline {16.13}IsAlgebraicElement}{395}
\contentsline {section}{\numberline {16.14}Algebraic extensions of the Rationals}{395}
\contentsline {section}{\numberline {16.15}DefectApproximation}{396}
\contentsline {section}{\numberline {16.16}GaloisType}{396}
\contentsline {section}{\numberline {16.17}ProbabilityShapes}{396}
\contentsline {section}{\numberline {16.18}DecomPoly}{396}
\contentsline {chapter}{\numberline {17}Unknowns}{399}
\contentsline {section}{\numberline {17.1}Unknown}{400}
\contentsline {section}{\numberline {17.2}IsUnknown}{400}
\contentsline {section}{\numberline {17.3}Comparisons of Unknowns}{401}
\contentsline {section}{\numberline {17.4}Operations for Unknowns}{401}
\contentsline {chapter}{\numberline {18}Finite Fields}{403}
\contentsline {section}{\numberline {18.1}Finite Field Elements}{403}
\contentsline {section}{\numberline {18.2}Comparisons of Finite Field Elements}{404}
\contentsline {section}{\numberline {18.3}Operations for Finite Field Elements}{405}
\contentsline {section}{\numberline {18.4}IsFFE}{406}
\contentsline {section}{\numberline {18.5}CharFFE}{406}
\contentsline {section}{\numberline {18.6}DegreeFFE}{407}
\contentsline {section}{\numberline {18.7}OrderFFE}{407}
\contentsline {section}{\numberline {18.8}IntFFE}{407}
\contentsline {section}{\numberline {18.9}LogFFE}{408}
\contentsline {section}{\numberline {18.10}GaloisField}{408}
\contentsline {section}{\numberline {18.11}FrobeniusAutomorphism}{409}
\contentsline {section}{\numberline {18.12}Set Functions for Finite Fields}{409}
\contentsline {section}{\numberline {18.13}Field Functions for Finite Fields}{410}
\contentsline {chapter}{\numberline {19}Polynomials}{411}
\contentsline {section}{\numberline {19.1}Multivariate Polynomials}{413}
\contentsline {section}{\numberline {19.2}Indeterminate}{413}
\contentsline {section}{\numberline {19.3}Polynomial}{414}
\contentsline {section}{\numberline {19.4}IsPolynomial}{414}
\contentsline {section}{\numberline {19.5}Comparisons of Polynomials}{414}
\contentsline {section}{\numberline {19.6}Operations for Polynomials}{415}
\contentsline {section}{\numberline {19.7}Degree}{417}
\contentsline {section}{\numberline {19.8}LeadingCoefficient}{418}
\contentsline {section}{\numberline {19.9}Value}{418}
\contentsline {section}{\numberline {19.10}Derivative}{418}
\contentsline {section}{\numberline {19.11}InterpolatedPolynomial}{419}
\contentsline {section}{\numberline {19.12}ConwayPolynomial}{419}
\contentsline {section}{\numberline {19.13}CyclotomicPolynomial}{419}
\contentsline {section}{\numberline {19.14}PolynomialRing}{420}
\contentsline {section}{\numberline {19.15}IsPolynomialRing}{420}
\contentsline {section}{\numberline {19.16}LaurentPolynomialRing}{420}
\contentsline {section}{\numberline {19.17}IsLaurentPolynomialRing}{421}
\contentsline {section}{\numberline {19.18}Ring Functions for Polynomial Rings}{421}
\contentsline {section}{\numberline {19.19}Ring Functions for Laurent Polynomial Rings}{423}
\contentsline {chapter}{\numberline {20}Permutations}{425}
\contentsline {section}{\numberline {20.1}Comparisons of Permutations}{426}
\contentsline {section}{\numberline {20.2}Operations for Permutations}{426}
\contentsline {section}{\numberline {20.3}IsPerm}{427}
\contentsline {section}{\numberline {20.4}LargestMovedPointPerm}{427}
\contentsline {section}{\numberline {20.5}SmallestMovedPointPerm}{428}
\contentsline {section}{\numberline {20.6}SignPerm}{428}
\contentsline {section}{\numberline {20.7}SmallestGeneratorPerm}{428}
\contentsline {section}{\numberline {20.8}ListPerm}{428}
\contentsline {section}{\numberline {20.9}PermList}{429}
\contentsline {section}{\numberline {20.10}RestrictedPerm}{429}
\contentsline {section}{\numberline {20.11}MappingPermListList}{429}
\contentsline {chapter}{\numberline {21}Permutation Groups}{431}
\contentsline {section}{\numberline {21.1}IsPermGroup}{431}
\contentsline {section}{\numberline {21.2}PermGroupOps.MovedPoints}{432}
\contentsline {section}{\numberline {21.3}PermGroupOps.SmallestMovedPoint}{432}
\contentsline {section}{\numberline {21.4}PermGroupOps.LargestMovedPoint}{432}
\contentsline {section}{\numberline {21.5}PermGroupOps.NrMovedPoints}{432}
\contentsline {section}{\numberline {21.6}Stabilizer Chains}{433}
\contentsline {section}{\numberline {21.7}StabChain}{434}
\contentsline {section}{\numberline {21.8}MakeStabChain}{435}
\contentsline {section}{\numberline {21.9}ExtendStabChain}{436}
\contentsline {section}{\numberline {21.10}ReduceStabChain}{436}
\contentsline {section}{\numberline {21.11}MakeStabChainStrongGenerators}{436}
\contentsline {section}{\numberline {21.12}Base for Permutation Groups}{437}
\contentsline {section}{\numberline {21.13}PermGroupOps.Indices}{437}
\contentsline {section}{\numberline {21.14}PermGroupOps.StrongGenerators}{437}
\contentsline {section}{\numberline {21.15}ListStabChain}{438}
\contentsline {section}{\numberline {21.16}PermGroupOps.ElementProperty}{438}
\contentsline {section}{\numberline {21.17}PermGroupOps.SubgroupProperty}{439}
\contentsline {section}{\numberline {21.18}CentralCompositionSeriesPPermGroup}{440}
\contentsline {section}{\numberline {21.19}PermGroupOps.PgGroup}{440}
\contentsline {section}{\numberline {21.20}Set Functions for Permutation Groups}{440}
\contentsline {section}{\numberline {21.21}Group Functions for Permutation Groups}{441}
\contentsline {section}{\numberline {21.22}Operations of Permutation Groups}{445}
\contentsline {section}{\numberline {21.23}Homomorphisms for Permutation Groups}{446}
\contentsline {section}{\numberline {21.24}Random Methods for Permutation Groups}{448}
\contentsline {section}{\numberline {21.25}Permutation Group Records}{449}
\contentsline {chapter}{\numberline {22}Words in Abstract Generators}{451}
\contentsline {section}{\numberline {22.1}AbstractGenerator}{452}
\contentsline {section}{\numberline {22.2}AbstractGenerators}{452}
\contentsline {section}{\numberline {22.3}Comparisons of Words}{453}
\contentsline {section}{\numberline {22.4}Operations for Words}{453}
\contentsline {section}{\numberline {22.5}IsWord}{454}
\contentsline {section}{\numberline {22.6}LengthWord}{455}
\contentsline {section}{\numberline {22.7}ExponentSumWord}{455}
\contentsline {section}{\numberline {22.8}Subword}{455}
\contentsline {section}{\numberline {22.9}PositionWord}{456}
\contentsline {section}{\numberline {22.10}SubstitutedWord}{456}
\contentsline {section}{\numberline {22.11}EliminatedWord}{456}
\contentsline {section}{\numberline {22.12}MappedWord}{457}
\contentsline {chapter}{\numberline {23}Finitely Presented Groups}{459}
\contentsline {section}{\numberline {23.1}FreeGroup}{460}
\contentsline {section}{\numberline {23.2}Set Functions for Finitely Presented Groups}{460}
\contentsline {section}{\numberline {23.3}Group Functions for Finitely Presented Groups}{461}
\contentsline {section}{\numberline {23.4}CosetTableFpGroup}{464}
\contentsline {section}{\numberline {23.5}OperationCosetsFpGroup}{465}
\contentsline {section}{\numberline {23.6}IsIdenticalPresentationFpGroup}{465}
\contentsline {section}{\numberline {23.7}LowIndexSubgroupsFpGroup}{466}
\contentsline {section}{\numberline {23.8}Presentation Records}{466}
\contentsline {section}{\numberline {23.9}Changing Presentations}{469}
\contentsline {section}{\numberline {23.10}Subgroup Presentations}{470}
\contentsline {section}{\numberline {23.11}SimplifiedFpGroup}{474}
\contentsline {section}{\numberline {23.12}Tietze Transformations}{474}
\contentsline {section}{\numberline {23.13}DecodeTree}{485}
\contentsline {chapter}{\numberline {24}Words in Finite Polycyclic Groups}{489}
\contentsline {section}{\numberline {24.1}More about Ag Words}{489}
\contentsline {section}{\numberline {24.2}Ag Word Comparisons}{490}
\contentsline {section}{\numberline {24.3}CentralWeight}{491}
\contentsline {section}{\numberline {24.4}CompositionLength}{491}
\contentsline {section}{\numberline {24.5}Depth}{491}
\contentsline {section}{\numberline {24.6}IsAgWord}{492}
\contentsline {section}{\numberline {24.7}LeadingExponent}{492}
\contentsline {section}{\numberline {24.8}RelativeOrder}{492}
\contentsline {section}{\numberline {24.9}CanonicalAgWord}{493}
\contentsline {section}{\numberline {24.10}DifferenceAgWord}{493}
\contentsline {section}{\numberline {24.11}ReducedAgWord}{494}
\contentsline {section}{\numberline {24.12}SiftedAgWord}{494}
\contentsline {section}{\numberline {24.13}SumAgWord}{494}
\contentsline {section}{\numberline {24.14}ExponentAgWord}{495}
\contentsline {section}{\numberline {24.15}ExponentsAgWord}{495}
\contentsline {chapter}{\numberline {25}Finite Polycyclic Groups}{497}
\contentsline {section}{\numberline {25.1}More about Ag Groups}{497}
\contentsline {section}{\numberline {25.2}Construction of Ag Groups}{498}
\contentsline {section}{\numberline {25.3}Ag Group Operations}{498}
\contentsline {section}{\numberline {25.4}Ag Group Records}{499}
\contentsline {section}{\numberline {25.5}Set Functions for Ag Groups}{499}
\contentsline {section}{\numberline {25.6}Elements for Ag Groups}{500}
\contentsline {section}{\numberline {25.7}Intersection for Ag Groups}{500}
\contentsline {section}{\numberline {25.8}Size for Ag Groups}{500}
\contentsline {section}{\numberline {25.9}Group Functions for Ag Groups}{501}
\contentsline {section}{\numberline {25.10}AsGroup for Ag Groups}{504}
\contentsline {section}{\numberline {25.11}Group for Ag Groups}{505}
\contentsline {section}{\numberline {25.12}CommutatorSubgroup for Ag Groups}{505}
\contentsline {section}{\numberline {25.13}Normalizer for Ag Groups}{505}
\contentsline {section}{\numberline {25.14}AbelianInvariants for Ag Groups}{505}
\contentsline {section}{\numberline {25.15}IsCyclic for Ag Groups}{506}
\contentsline {section}{\numberline {25.16}IsNormal for Ag Groups}{506}
\contentsline {section}{\numberline {25.17}IsSubgroup for Ag Groups}{506}
\contentsline {section}{\numberline {25.18}Stabilizer for Ag Groups}{506}
\contentsline {section}{\numberline {25.19}CyclicGroup for Ag Groups}{506}
\contentsline {section}{\numberline {25.20}ElementaryAbelianGroup for Ag Groups}{507}
\contentsline {section}{\numberline {25.21}DirectProduct for Ag Groups}{507}
\contentsline {section}{\numberline {25.22}WreathProduct for Ag Groups}{508}
\contentsline {section}{\numberline {25.23}RightCoset for Ag Groups}{508}
\contentsline {section}{\numberline {25.24}FpGroup for Ag Groups}{509}
\contentsline {section}{\numberline {25.25}Ag Group Functions}{509}
\contentsline {section}{\numberline {25.26}AgGroup}{509}
\contentsline {section}{\numberline {25.27}IsAgGroup}{510}
\contentsline {section}{\numberline {25.28}AgGroupFpGroup}{510}
\contentsline {section}{\numberline {25.29}IsConsistent}{510}
\contentsline {section}{\numberline {25.30}IsElementaryAbelianAgSeries}{511}
\contentsline {section}{\numberline {25.31}MatGroupAgGroup}{511}
\contentsline {section}{\numberline {25.32}PermGroupAgGroup}{512}
\contentsline {section}{\numberline {25.33}RefinedAgSeries}{512}
\contentsline {section}{\numberline {25.34}ChangeCollector}{512}
\contentsline {section}{\numberline {25.35}The Prime Quotient Algorithm}{513}
\contentsline {section}{\numberline {25.36}PQuotient}{513}
\contentsline {section}{\numberline {25.37}Save}{515}
\contentsline {section}{\numberline {25.38}PQp}{516}
\contentsline {section}{\numberline {25.39}InitPQp}{516}
\contentsline {section}{\numberline {25.40}FirstClassPQp}{516}
\contentsline {section}{\numberline {25.41}NextClassPQp}{516}
\contentsline {section}{\numberline {25.42}Weight}{517}
\contentsline {section}{\numberline {25.43}Factorization for PQp}{517}
\contentsline {section}{\numberline {25.44}The Solvable Quotient Algorithm}{517}
\contentsline {section}{\numberline {25.45}SolvableQuotient}{517}
\contentsline {section}{\numberline {25.46}InitSQ}{519}
\contentsline {section}{\numberline {25.47}ModulesSQ}{519}
\contentsline {section}{\numberline {25.48}NextModuleSQ}{520}
\contentsline {section}{\numberline {25.49}Generating Systems of Ag Groups}{520}
\contentsline {section}{\numberline {25.50}AgSubgroup}{521}
\contentsline {section}{\numberline {25.51}Cgs}{521}
\contentsline {section}{\numberline {25.52}Igs}{522}
\contentsline {section}{\numberline {25.53}IsNormalized}{522}
\contentsline {section}{\numberline {25.54}Normalize}{522}
\contentsline {section}{\numberline {25.55}Normalized}{522}
\contentsline {section}{\numberline {25.56}MergedCgs}{522}
\contentsline {section}{\numberline {25.57}MergedIgs}{523}
\contentsline {section}{\numberline {25.58}Factor Groups of Ag Groups}{523}
\contentsline {section}{\numberline {25.59}FactorGroup for AgGroups}{524}
\contentsline {section}{\numberline {25.60}CollectorlessFactorGroup}{524}
\contentsline {section}{\numberline {25.61}FactorArg}{524}
\contentsline {section}{\numberline {25.62}Subgroups and Properties of Ag Groups}{525}
\contentsline {section}{\numberline {25.63}CompositionSubgroup}{525}
\contentsline {section}{\numberline {25.64}HallSubgroup}{526}
\contentsline {section}{\numberline {25.65}PRump}{526}
\contentsline {section}{\numberline {25.66}RefinedSubnormalSeries}{526}
\contentsline {section}{\numberline {25.67}SylowComplements}{527}
\contentsline {section}{\numberline {25.68}SylowSystem}{527}
\contentsline {section}{\numberline {25.69}SystemNormalizer}{528}
\contentsline {section}{\numberline {25.70}MinimalGeneratingSet}{529}
\contentsline {section}{\numberline {25.71}IsElementAgSeries}{529}
\contentsline {section}{\numberline {25.72}IsPNilpotent}{529}
\contentsline {section}{\numberline {25.73}NumberConjugacyClasses}{529}
\contentsline {section}{\numberline {25.74}Exponents}{530}
\contentsline {section}{\numberline {25.75}FactorsAgGroup}{530}
\contentsline {section}{\numberline {25.76}MaximalElement}{531}
\contentsline {section}{\numberline {25.77}Orbitalgorithms of Ag Groups}{531}
\contentsline {section}{\numberline {25.78}AffineOperation}{531}
\contentsline {section}{\numberline {25.79}AgOrbitStabilizer}{532}
\contentsline {section}{\numberline {25.80}LinearOperation}{532}
\contentsline {section}{\numberline {25.81}Intersections of Ag Groups}{533}
\contentsline {section}{\numberline {25.82}ExtendedIntersectionSumAgGroup}{533}
\contentsline {section}{\numberline {25.83}IntersectionSumAgGroup}{534}
\contentsline {section}{\numberline {25.84}SumAgGroup}{535}
\contentsline {section}{\numberline {25.85}SumFactorizationFunctionAgGroup}{535}
\contentsline {section}{\numberline {25.86}One Cohomology Group}{536}
\contentsline {section}{\numberline {25.87}OneCoboundaries}{536}
\contentsline {section}{\numberline {25.88}OneCocycles}{537}
\contentsline {section}{\numberline {25.89}Complements}{539}
\contentsline {section}{\numberline {25.90}Complement}{539}
\contentsline {section}{\numberline {25.91}Complementclasses}{539}
\contentsline {section}{\numberline {25.92}CoprimeComplement}{540}
\contentsline {section}{\numberline {25.93}ComplementConjugatingAgWord}{540}
\contentsline {section}{\numberline {25.94}HallConjugatingWordAgGroup}{541}
\contentsline {section}{\numberline {25.95}Example, normal closure}{541}
\contentsline {chapter}{\numberline {26}Special Ag Groups}{545}
\contentsline {section}{\numberline {26.1}More about Special Ag Groups}{545}
\contentsline {section}{\numberline {26.2}Construction of Special Ag Groups}{547}
\contentsline {section}{\numberline {26.3}Restricted Special Ag Groups}{547}
\contentsline {section}{\numberline {26.4}Special Ag Group Records}{548}
\contentsline {section}{\numberline {26.5}MatGroupSagGroup}{550}
\contentsline {section}{\numberline {26.6}DualMatGroupSagGroup}{550}
\contentsline {section}{\numberline {26.7}Ag Group Functions for Special Ag Groups}{551}
\contentsline {chapter}{\numberline {27}Lists}{553}
\contentsline {section}{\numberline {27.1}IsList}{554}
\contentsline {section}{\numberline {27.2}List}{554}
\contentsline {section}{\numberline {27.3}List Elements}{555}
\contentsline {section}{\numberline {27.4}Length}{556}
\contentsline {section}{\numberline {27.5}List Assignment}{556}
\contentsline {section}{\numberline {27.6}Add}{558}
\contentsline {section}{\numberline {27.7}Append}{558}
\contentsline {section}{\numberline {27.8}Identical Lists}{559}
\contentsline {section}{\numberline {27.9}IsIdentical}{560}
\contentsline {section}{\numberline {27.10}Enlarging Lists}{561}
\contentsline {section}{\numberline {27.11}Comparisons of Lists}{562}
\contentsline {section}{\numberline {27.12}Operations for Lists}{562}
\contentsline {section}{\numberline {27.13}In}{563}
\contentsline {section}{\numberline {27.14}Position}{563}
\contentsline {section}{\numberline {27.15}PositionSorted}{564}
\contentsline {section}{\numberline {27.16}PositionProperty}{565}
\contentsline {section}{\numberline {27.17}Concatenation}{565}
\contentsline {section}{\numberline {27.18}Flat}{565}
\contentsline {section}{\numberline {27.19}Reversed}{566}
\contentsline {section}{\numberline {27.20}Sublist}{566}
\contentsline {section}{\numberline {27.21}Cartesian}{566}
\contentsline {section}{\numberline {27.22}Number}{567}
\contentsline {section}{\numberline {27.23}Collected}{567}
\contentsline {section}{\numberline {27.24}Filtered}{568}
\contentsline {section}{\numberline {27.25}ForAll}{568}
\contentsline {section}{\numberline {27.26}ForAny}{569}
\contentsline {section}{\numberline {27.27}First}{569}
\contentsline {section}{\numberline {27.28}Sort}{569}
\contentsline {section}{\numberline {27.29}SortParallel}{570}
\contentsline {section}{\numberline {27.30}Sortex}{570}
\contentsline {section}{\numberline {27.31}Permuted}{570}
\contentsline {section}{\numberline {27.32}Product}{571}
\contentsline {section}{\numberline {27.33}Sum}{571}
\contentsline {section}{\numberline {27.34}Maximum}{571}
\contentsline {section}{\numberline {27.35}Minimum}{572}
\contentsline {section}{\numberline {27.36}Iterated}{572}
\contentsline {section}{\numberline {27.37}RandomList}{572}
\contentsline {chapter}{\numberline {28}Sets}{575}
\contentsline {section}{\numberline {28.1}IsSet}{576}
\contentsline {section}{\numberline {28.2}Set}{576}
\contentsline {section}{\numberline {28.3}IsEqualSet}{576}
\contentsline {section}{\numberline {28.4}AddSet}{577}
\contentsline {section}{\numberline {28.5}RemoveSet}{577}
\contentsline {section}{\numberline {28.6}UniteSet}{577}
\contentsline {section}{\numberline {28.7}IntersectSet}{578}
\contentsline {section}{\numberline {28.8}SubtractSet}{578}
\contentsline {section}{\numberline {28.9}Set Functions for Sets}{578}
\contentsline {section}{\numberline {28.10}More about Sets}{579}
\contentsline {chapter}{\numberline {29}Boolean Lists}{581}
\contentsline {section}{\numberline {29.1}BlistList}{581}
\contentsline {section}{\numberline {29.2}ListBlist}{582}
\contentsline {section}{\numberline {29.3}IsBlist}{582}
\contentsline {section}{\numberline {29.4}SizeBlist}{582}
\contentsline {section}{\numberline {29.5}IsSubsetBlist}{583}
\contentsline {section}{\numberline {29.6}UnionBlist}{583}
\contentsline {section}{\numberline {29.7}IntersectionBlist}{583}
\contentsline {section}{\numberline {29.8}DifferenceBlist}{584}
\contentsline {section}{\numberline {29.9}UniteBlist}{584}
\contentsline {section}{\numberline {29.10}IntersectBlist}{584}
\contentsline {section}{\numberline {29.11}SubtractBlist}{584}
\contentsline {section}{\numberline {29.12}More about Boolean Lists}{585}
\contentsline {chapter}{\numberline {30}Strings and Characters}{587}
\contentsline {section}{\numberline {30.1}String}{589}
\contentsline {section}{\numberline {30.2}ConcatenationString}{589}
\contentsline {section}{\numberline {30.3}SubString}{590}
\contentsline {section}{\numberline {30.4}Comparisons of Strings}{590}
\contentsline {section}{\numberline {30.5}IsString}{591}
\contentsline {section}{\numberline {30.6}LengthString}{591}
\contentsline {chapter}{\numberline {31}Ranges}{593}
\contentsline {section}{\numberline {31.1}IsRange}{594}
\contentsline {section}{\numberline {31.2}More about Ranges}{594}
\contentsline {chapter}{\numberline {32}Vectors}{597}
\contentsline {section}{\numberline {32.1}Operations for Vectors}{598}
\contentsline {section}{\numberline {32.2}IsVector}{599}
\contentsline {section}{\numberline {32.3}NormedVector}{599}
\contentsline {section}{\numberline {32.4}More about Vectors}{599}
\contentsline {chapter}{\numberline {33}Row Spaces}{601}
\contentsline {section}{\numberline {33.1}More about Row Spaces}{601}
\contentsline {section}{\numberline {33.2}Row Space Bases}{602}
\contentsline {section}{\numberline {33.3}Row Space Cosets}{602}
\contentsline {section}{\numberline {33.4}Quotient Spaces}{603}
\contentsline {section}{\numberline {33.5}Subspaces and Parent Spaces}{603}
\contentsline {section}{\numberline {33.6}RowSpace}{604}
\contentsline {section}{\numberline {33.7}Operations for Row Spaces}{604}
\contentsline {section}{\numberline {33.8}Functions for Row Spaces}{605}
\contentsline {section}{\numberline {33.9}IsRowSpace}{606}
\contentsline {section}{\numberline {33.10}Subspace}{606}
\contentsline {section}{\numberline {33.11}AsSubspace}{606}
\contentsline {section}{\numberline {33.12}AsSpace}{607}
\contentsline {section}{\numberline {33.13}NormedVectors}{607}
\contentsline {section}{\numberline {33.14}Coefficients for Row Space Bases}{607}
\contentsline {section}{\numberline {33.15}SiftedVector}{607}
\contentsline {section}{\numberline {33.16}Basis}{608}
\contentsline {section}{\numberline {33.17}CanonicalBasis}{608}
\contentsline {section}{\numberline {33.18}SemiEchelonBasis}{609}
\contentsline {section}{\numberline {33.19}IsSemiEchelonBasis}{609}
\contentsline {section}{\numberline {33.20}NumberVector}{610}
\contentsline {section}{\numberline {33.21}ElementRowSpace}{610}
\contentsline {section}{\numberline {33.22}Operations for Row Space Cosets}{610}
\contentsline {section}{\numberline {33.23}Functions for Row Space Cosets}{611}
\contentsline {section}{\numberline {33.24}IsSpaceCoset}{611}
\contentsline {section}{\numberline {33.25}Operations for Quotient Spaces}{612}
\contentsline {section}{\numberline {33.26}Functions for Quotient Spaces}{612}
\contentsline {section}{\numberline {33.27}Row Space Records}{612}
\contentsline {section}{\numberline {33.28}Row Space Basis Records}{613}
\contentsline {section}{\numberline {33.29}Row Space Coset Records}{613}
\contentsline {section}{\numberline {33.30}Quotient Space Records}{614}
\contentsline {chapter}{\numberline {34}Matrices}{615}
\contentsline {section}{\numberline {34.1}Operations for Matrices}{615}
\contentsline {section}{\numberline {34.2}IsMat}{617}
\contentsline {section}{\numberline {34.3}IdentityMat}{618}
\contentsline {section}{\numberline {34.4}NullMat}{618}
\contentsline {section}{\numberline {34.5}TransposedMat}{618}
\contentsline {section}{\numberline {34.6}KroneckerProduct}{619}
\contentsline {section}{\numberline {34.7}DimensionsMat}{619}
\contentsline {section}{\numberline {34.8}TraceMat}{619}
\contentsline {section}{\numberline {34.9}DeterminantMat}{619}
\contentsline {section}{\numberline {34.10}RankMat}{620}
\contentsline {section}{\numberline {34.11}OrderMat}{620}
\contentsline {section}{\numberline {34.12}TriangulizeMat}{620}
\contentsline {section}{\numberline {34.13}BaseMat}{621}
\contentsline {section}{\numberline {34.14}NullspaceMat}{621}
\contentsline {section}{\numberline {34.15}SolutionMat}{621}
\contentsline {section}{\numberline {34.16}DiagonalizeMat}{622}
\contentsline {section}{\numberline {34.17}ElementaryDivisorsMat}{622}
\contentsline {section}{\numberline {34.18}PrintArray}{622}
\contentsline {chapter}{\numberline {35}Matrix Rings}{623}
\contentsline {section}{\numberline {35.1}Set Functions for Matrix Rings}{623}
\contentsline {section}{\numberline {35.2}Ring Functions for Matrix Rings}{624}
\contentsline {chapter}{\numberline {36}Matrix Groups}{625}
\contentsline {section}{\numberline {36.1}Set Functions for Matrix Groups}{625}
\contentsline {section}{\numberline {36.2}Group Functions for Matrix Groups}{626}
\contentsline {section}{\numberline {36.3}Matrix Group Records}{627}
\contentsline {chapter}{\numberline {37}Group Libraries}{629}
\contentsline {section}{\numberline {37.1}The Basic Groups Library}{629}
\contentsline {section}{\numberline {37.2}Selection Functions}{632}
\contentsline {section}{\numberline {37.3}Example Functions}{633}
\contentsline {section}{\numberline {37.4}Extraction Functions}{634}
\contentsline {section}{\numberline {37.5}The Primitive Groups Library}{636}
\contentsline {section}{\numberline {37.6}The Transitive Groups Library}{638}
\contentsline {section}{\numberline {37.7}The Solvable Groups Library}{640}
\contentsline {section}{\numberline {37.8}The 2-Groups Library}{641}
\contentsline {section}{\numberline {37.9}The 3-Groups Library}{643}
\contentsline {section}{\numberline {37.10}The Irreducible Solvable Linear Groups Library}{645}
\contentsline {section}{\numberline {37.11}The Crystallographic Groups Library}{647}
\contentsline {chapter}{\numberline {38}Algebras}{661}
\contentsline {section}{\numberline {38.1}More about Algebras}{662}
\contentsline {section}{\numberline {38.2}Algebras and Unital Algebras}{662}
\contentsline {section}{\numberline {38.3}Parent Algebras and Subalgebras}{663}
\contentsline {section}{\numberline {38.4}Algebra}{664}
\contentsline {section}{\numberline {38.5}UnitalAlgebra}{664}
\contentsline {section}{\numberline {38.6}IsAlgebra}{665}
\contentsline {section}{\numberline {38.7}IsUnitalAlgebra}{665}
\contentsline {section}{\numberline {38.8}Subalgebra}{665}
\contentsline {section}{\numberline {38.9}UnitalSubalgebra}{666}
\contentsline {section}{\numberline {38.10}IsSubalgebra}{666}
\contentsline {section}{\numberline {38.11}AsAlgebra}{667}
\contentsline {section}{\numberline {38.12}AsUnitalAlgebra}{667}
\contentsline {section}{\numberline {38.13}AsSubalgebra}{667}
\contentsline {section}{\numberline {38.14}AsUnitalSubalgebra}{668}
\contentsline {section}{\numberline {38.15}Operations for Algebras}{668}
\contentsline {section}{\numberline {38.16}Zero and One for Algebras}{669}
\contentsline {section}{\numberline {38.17}Set Theoretic Functions for Algebras}{669}
\contentsline {section}{\numberline {38.18}Property Tests for Algebras}{670}
\contentsline {section}{\numberline {38.19}Vector Space Functions for Algebras}{670}
\contentsline {section}{\numberline {38.20}Algebra Functions for Algebras}{671}
\contentsline {section}{\numberline {38.21}TrivialSubalgebra}{672}
\contentsline {section}{\numberline {38.22}Operation for Algebras}{672}
\contentsline {section}{\numberline {38.23}OperationHomomorphism for Algebras}{673}
\contentsline {section}{\numberline {38.24}Algebra Homomorphisms}{673}
\contentsline {section}{\numberline {38.25}Mapping Functions for Algebra Homomorphisms}{673}
\contentsline {section}{\numberline {38.26}Algebra Elements}{674}
\contentsline {section}{\numberline {38.27}IsAlgebraElement}{675}
\contentsline {section}{\numberline {38.28}Algebra Records}{675}
\contentsline {section}{\numberline {38.29}FFList}{676}
\contentsline {chapter}{\numberline {39}Finitely Presented Algebras}{677}
\contentsline {section}{\numberline {39.1}More about Finitely Presented Algebras}{677}
\contentsline {section}{\numberline {39.2}FreeAlgebra}{678}
\contentsline {section}{\numberline {39.3}FpAlgebra}{679}
\contentsline {section}{\numberline {39.4}IsFpAlgebra}{679}
\contentsline {section}{\numberline {39.5}Operators for Finitely Presented Algebras}{680}
\contentsline {section}{\numberline {39.6}Functions for Finitely Presented Algebras}{680}
\contentsline {section}{\numberline {39.7}PrintDefinitionFpAlgebra}{681}
\contentsline {section}{\numberline {39.8}MappedExpression}{681}
\contentsline {section}{\numberline {39.9}Elements of Finitely Presented Algebras}{681}
\contentsline {section}{\numberline {39.10}ElementAlgebra}{683}
\contentsline {section}{\numberline {39.11}NumberAlgebraElement}{683}
\contentsline {chapter}{\numberline {40}Matrix Algebras}{685}
\contentsline {section}{\numberline {40.1}More about Matrix Algebras}{685}
\contentsline {section}{\numberline {40.2}Bases for Matrix Algebras}{686}
\contentsline {section}{\numberline {40.3}IsMatAlgebra}{686}
\contentsline {section}{\numberline {40.4}Zero and One for Matrix Algebras}{686}
\contentsline {section}{\numberline {40.5}Functions for Matrix Algebras}{686}
\contentsline {section}{\numberline {40.6}Algebra Functions for Matrix Algebras}{687}
\contentsline {section}{\numberline {40.7}RepresentativeOperation for Matrix Algebras}{687}
\contentsline {section}{\numberline {40.8}MatAlgebra}{687}
\contentsline {section}{\numberline {40.9}NullAlgebra}{688}
\contentsline {section}{\numberline {40.10}Fingerprint}{688}
\contentsline {section}{\numberline {40.11}NaturalModule}{689}
\contentsline {chapter}{\numberline {41}Modules}{691}
\contentsline {section}{\numberline {41.1}More about Modules}{691}
\contentsline {section}{\numberline {41.2}Row Modules}{692}
\contentsline {section}{\numberline {41.3}Free Modules}{692}
\contentsline {section}{\numberline {41.4}Module}{693}
\contentsline {section}{\numberline {41.5}Submodule}{693}
\contentsline {section}{\numberline {41.6}AsModule}{694}
\contentsline {section}{\numberline {41.7}AsSubmodule}{694}
\contentsline {section}{\numberline {41.8}AsSpace for Modules}{694}
\contentsline {section}{\numberline {41.9}IsModule}{694}
\contentsline {section}{\numberline {41.10}IsFreeModule}{695}
\contentsline {section}{\numberline {41.11}Operations for Row Modules}{695}
\contentsline {section}{\numberline {41.12}Functions for Row Modules}{696}
\contentsline {section}{\numberline {41.13}StandardBasis for Row Modules}{696}
\contentsline {section}{\numberline {41.14}IsEquivalent for Row Modules}{696}
\contentsline {section}{\numberline {41.15}IsIrreducible for Row Modules}{697}
\contentsline {section}{\numberline {41.16}FixedSubmodule}{697}
\contentsline {section}{\numberline {41.17}Module Homomorphisms}{697}
\contentsline {section}{\numberline {41.18}Row Module Records}{698}
\contentsline {section}{\numberline {41.19}Module Homomorphism Records}{699}
\contentsline {chapter}{\numberline {42}Mappings}{701}
\contentsline {section}{\numberline {42.1}IsGeneralMapping}{702}
\contentsline {section}{\numberline {42.2}IsMapping}{702}
\contentsline {section}{\numberline {42.3}IsInjective}{703}
\contentsline {section}{\numberline {42.4}IsSurjective}{703}
\contentsline {section}{\numberline {42.5}IsBijection}{704}
\contentsline {section}{\numberline {42.6}Comparisons of Mappings}{705}
\contentsline {section}{\numberline {42.7}Operations for Mappings}{706}
\contentsline {section}{\numberline {42.8}Image}{708}
\contentsline {section}{\numberline {42.9}Images}{710}
\contentsline {section}{\numberline {42.10}ImagesRepresentative}{711}
\contentsline {section}{\numberline {42.11}PreImage}{711}
\contentsline {section}{\numberline {42.12}PreImages}{713}
\contentsline {section}{\numberline {42.13}PreImagesRepresentative}{714}
\contentsline {section}{\numberline {42.14}CompositionMapping}{714}
\contentsline {section}{\numberline {42.15}PowerMapping}{715}
\contentsline {section}{\numberline {42.16}InverseMapping}{716}
\contentsline {section}{\numberline {42.17}IdentityMapping}{716}
\contentsline {section}{\numberline {42.18}MappingByFunction}{717}
\contentsline {section}{\numberline {42.19}Mapping Records}{717}
\contentsline {chapter}{\numberline {43}Homomorphisms}{719}
\contentsline {section}{\numberline {43.1}IsHomomorphism}{719}
\contentsline {section}{\numberline {43.2}IsMonomorphism}{720}
\contentsline {section}{\numberline {43.3}IsEpimorphism}{721}
\contentsline {section}{\numberline {43.4}IsIsomorphism}{721}
\contentsline {section}{\numberline {43.5}IsEndomorphism}{722}
\contentsline {section}{\numberline {43.6}IsAutomorphism}{722}
\contentsline {section}{\numberline {43.7}Kernel}{723}
\contentsline {chapter}{\numberline {44}Booleans}{725}
\contentsline {section}{\numberline {44.1}Comparisons of Booleans}{725}
\contentsline {section}{\numberline {44.2}Operations for Booleans}{726}
\contentsline {section}{\numberline {44.3}IsBool}{727}
\contentsline {chapter}{\numberline {45}Records}{729}
\contentsline {section}{\numberline {45.1}Accessing Record Elements}{730}
\contentsline {section}{\numberline {45.2}Record Assignment}{730}
\contentsline {section}{\numberline {45.3}Identical Records}{731}
\contentsline {section}{\numberline {45.4}Comparisons of Records}{733}
\contentsline {section}{\numberline {45.5}Operations for Records}{735}
\contentsline {section}{\numberline {45.6}In for Records}{736}
\contentsline {section}{\numberline {45.7}Printing of Records}{737}
\contentsline {section}{\numberline {45.8}IsRec}{738}
\contentsline {section}{\numberline {45.9}IsBound}{738}
\contentsline {section}{\numberline {45.10}Unbind}{739}
\contentsline {section}{\numberline {45.11}Copy}{739}
\contentsline {section}{\numberline {45.12}ShallowCopy}{740}
\contentsline {section}{\numberline {45.13}RecFields}{741}
\contentsline {chapter}{\numberline {46}Combinatorics}{743}
\contentsline {section}{\numberline {46.1}Factorial}{743}
\contentsline {section}{\numberline {46.2}Binomial}{744}
\contentsline {section}{\numberline {46.3}Bell}{744}
\contentsline {section}{\numberline {46.4}Stirling1}{745}
\contentsline {section}{\numberline {46.5}Stirling2}{745}
\contentsline {section}{\numberline {46.6}Combinations}{746}
\contentsline {section}{\numberline {46.7}Arrangements}{746}
\contentsline {section}{\numberline {46.8}UnorderedTuples}{747}
\contentsline {section}{\numberline {46.9}Tuples}{748}
\contentsline {section}{\numberline {46.10}PermutationsList}{748}
\contentsline {section}{\numberline {46.11}Derangements}{749}
\contentsline {section}{\numberline {46.12}Permanent}{749}
\contentsline {section}{\numberline {46.13}PartitionsSet}{750}
\contentsline {section}{\numberline {46.14}Partitions}{750}
\contentsline {section}{\numberline {46.15}OrderedPartitions}{751}
\contentsline {section}{\numberline {46.16}RestrictedPartitions}{752}
\contentsline {section}{\numberline {46.17}SignPartition}{753}
\contentsline {section}{\numberline {46.18}AssociatedPartition}{753}
\contentsline {section}{\numberline {46.19}PowerPartition}{753}
\contentsline {section}{\numberline {46.20}PartitionTuples}{753}
\contentsline {section}{\numberline {46.21}Fibonacci}{754}
\contentsline {section}{\numberline {46.22}Lucas}{754}
\contentsline {section}{\numberline {46.23}Bernoulli}{755}
\contentsline {chapter}{\numberline {47}Tables of Marks}{757}
\contentsline {section}{\numberline {47.1}More about Tables of Marks}{757}
\contentsline {section}{\numberline {47.2}Table of Marks Records}{758}
\contentsline {section}{\numberline {47.3}The Library of Tables of Marks}{758}
\contentsline {section}{\numberline {47.4}TableOfMarks}{759}
\contentsline {section}{\numberline {47.5}Marks}{760}
\contentsline {section}{\numberline {47.6}NrSubs}{760}
\contentsline {section}{\numberline {47.7}WeightsTom}{760}
\contentsline {section}{\numberline {47.8}MatTom}{761}
\contentsline {section}{\numberline {47.9}TomMat}{761}
\contentsline {section}{\numberline {47.10}DecomposedFixedPointVector}{761}
\contentsline {section}{\numberline {47.11}TestTom}{762}
\contentsline {section}{\numberline {47.12}DisplayTom}{762}
\contentsline {section}{\numberline {47.13}NormalizerTom}{763}
\contentsline {section}{\numberline {47.14}IntersectionsTom}{763}
\contentsline {section}{\numberline {47.15}IsCyclicTom}{764}
\contentsline {section}{\numberline {47.16}FusionCharTableTom}{764}
\contentsline {section}{\numberline {47.17}PermCharsTom}{764}
\contentsline {section}{\numberline {47.18}MoebiusTom}{764}
\contentsline {section}{\numberline {47.19}CyclicExtensionsTom}{765}
\contentsline {section}{\numberline {47.20}IdempotentsTom}{765}
\contentsline {section}{\numberline {47.21}ClassTypesTom}{765}
\contentsline {section}{\numberline {47.22}ClassNamesTom}{765}
\contentsline {section}{\numberline {47.23}TomCyclic}{766}
\contentsline {section}{\numberline {47.24}TomDihedral}{766}
\contentsline {section}{\numberline {47.25}TomFrobenius}{767}
\contentsline {chapter}{\numberline {48}Character Tables}{769}
\contentsline {section}{\numberline {48.1}Some Notes on Character Theory in GAP}{769}
\contentsline {section}{\numberline {48.2}Character Table Records}{771}
\contentsline {section}{\numberline {48.3}Brauer Table Records}{775}
\contentsline {section}{\numberline {48.4}IsCharTable}{777}
\contentsline {section}{\numberline {48.5}PrintCharTable}{777}
\contentsline {section}{\numberline {48.6}TestCharTable}{777}
\contentsline {section}{\numberline {48.7}Operations Records for Character Tables}{778}
\contentsline {section}{\numberline {48.8}Functions for Character Tables}{778}
\contentsline {section}{\numberline {48.9}Operators for Character Tables}{779}
\contentsline {section}{\numberline {48.10}Conventions for Character Tables}{779}
\contentsline {section}{\numberline {48.11}Getting Character Tables}{780}
\contentsline {section}{\numberline {48.12}CharTable}{780}
\contentsline {section}{\numberline {48.13}Advanced Methods for Dixon Schneider Calculations}{784}
\contentsline {section}{\numberline {48.14}An Example of Advanced Dixon Schneider Calculations}{785}
\contentsline {section}{\numberline {48.15}CharTableFactorGroup}{787}
\contentsline {section}{\numberline {48.16}CharTableNormalSubgroup}{788}
\contentsline {section}{\numberline {48.17}CharTableDirectProduct}{788}
\contentsline {section}{\numberline {48.18}CharTableWreathSymmetric}{789}
\contentsline {section}{\numberline {48.19}CharTableRegular}{791}
\contentsline {section}{\numberline {48.20}CharTableIsoclinic}{791}
\contentsline {section}{\numberline {48.21}CharTableSplitClasses}{792}
\contentsline {section}{\numberline {48.22}CharTableCollapsedClasses}{793}
\contentsline {section}{\numberline {48.23}CharDegAgGroup}{794}
\contentsline {section}{\numberline {48.24}CharTableSSGroup}{794}
\contentsline {section}{\numberline {48.25}MatRepresentationsPGroup}{795}
\contentsline {section}{\numberline {48.26}CharTablePGroup}{796}
\contentsline {section}{\numberline {48.27}InitClassesCharTable}{796}
\contentsline {section}{\numberline {48.28}InverseClassesCharTable}{797}
\contentsline {section}{\numberline {48.29}ClassNamesCharTable}{797}
\contentsline {section}{\numberline {48.30}ClassMultCoeffCharTable}{797}
\contentsline {section}{\numberline {48.31}MatClassMultCoeffsCharTable}{798}
\contentsline {section}{\numberline {48.32}ClassStructureCharTable}{798}
\contentsline {section}{\numberline {48.33}RealClassesCharTable}{799}
\contentsline {section}{\numberline {48.34}ClassOrbitCharTable}{799}
\contentsline {section}{\numberline {48.35}ClassRootsCharTable}{799}
\contentsline {section}{\numberline {48.36}NrPolyhedralSubgroups}{799}
\contentsline {section}{\numberline {48.37}DisplayCharTable}{800}
\contentsline {section}{\numberline {48.38}SortCharactersCharTable}{801}
\contentsline {section}{\numberline {48.39}SortClassesCharTable}{803}
\contentsline {section}{\numberline {48.40}SortCharTable}{804}
\contentsline {section}{\numberline {48.41}MatAutomorphisms}{805}
\contentsline {section}{\numberline {48.42}TableAutomorphisms}{806}
\contentsline {section}{\numberline {48.43}TransformingPermutations}{806}
\contentsline {section}{\numberline {48.44}TransformingPermutationsCharTables}{806}
\contentsline {section}{\numberline {48.45}GetFusionMap}{807}
\contentsline {section}{\numberline {48.46}StoreFusion}{808}
\contentsline {section}{\numberline {48.47}FusionConjugacyClasses}{808}
\contentsline {section}{\numberline {48.48}MAKElb11}{809}
\contentsline {section}{\numberline {48.49}ScanMOC}{809}
\contentsline {section}{\numberline {48.50}MOCChars}{809}
\contentsline {section}{\numberline {48.51}GAPChars}{809}
\contentsline {section}{\numberline {48.52}MOCTable}{809}
\contentsline {section}{\numberline {48.53}PrintToMOC}{811}
\contentsline {section}{\numberline {48.54}PrintToCAS}{811}
\contentsline {chapter}{\numberline {49}Generic Character Tables}{813}
\contentsline {section}{\numberline {49.1}More about Generic Character Tables}{813}
\contentsline {section}{\numberline {49.2}Examples of Generic Character Tables}{814}
\contentsline {section}{\numberline {49.3}CharTableSpecialized}{816}
\contentsline {chapter}{\numberline {50}Characters}{817}
\contentsline {section}{\numberline {50.1}ScalarProduct}{817}
\contentsline {section}{\numberline {50.2}MatScalarProducts}{818}
\contentsline {section}{\numberline {50.3}Decomposition}{818}
\contentsline {section}{\numberline {50.4}Subroutines of Decomposition}{819}
\contentsline {section}{\numberline {50.5}KernelChar}{820}
\contentsline {section}{\numberline {50.6}PrimeBlocks}{820}
\contentsline {section}{\numberline {50.7}Indicator}{821}
\contentsline {section}{\numberline {50.8}Eigenvalues}{821}
\contentsline {section}{\numberline {50.9}MolienSeries}{822}
\contentsline {section}{\numberline {50.10}Reduced}{822}
\contentsline {section}{\numberline {50.11}ReducedOrdinary}{823}
\contentsline {section}{\numberline {50.12}Tensored}{823}
\contentsline {section}{\numberline {50.13}Symmetrisations}{824}
\contentsline {section}{\numberline {50.14}SymmetricParts}{824}
\contentsline {section}{\numberline {50.15}AntiSymmetricParts}{825}
\contentsline {section}{\numberline {50.16}MinusCharacter}{825}
\contentsline {section}{\numberline {50.17}OrthogonalComponents}{825}
\contentsline {section}{\numberline {50.18}SymplecticComponents}{826}
\contentsline {section}{\numberline {50.19}IrreducibleDifferences}{826}
\contentsline {section}{\numberline {50.20}Restricted}{827}
\contentsline {section}{\numberline {50.21}Inflated}{827}
\contentsline {section}{\numberline {50.22}Induced}{828}
\contentsline {section}{\numberline {50.23}InducedCyclic}{828}
\contentsline {section}{\numberline {50.24}CollapsedMat}{829}
\contentsline {section}{\numberline {50.25}Power}{829}
\contentsline {section}{\numberline {50.26}Permutation Character Candidates}{830}
\contentsline {section}{\numberline {50.27}IsPermChar}{830}
\contentsline {section}{\numberline {50.28}PermCharInfo}{830}
\contentsline {section}{\numberline {50.29}Inequalities}{831}
\contentsline {section}{\numberline {50.30}PermBounds}{832}
\contentsline {section}{\numberline {50.31}PermChars}{832}
\contentsline {section}{\numberline {50.32}Faithful Permutation Characters}{833}
\contentsline {section}{\numberline {50.33}LLLReducedBasis}{834}
\contentsline {section}{\numberline {50.34}LLLReducedGramMat}{835}
\contentsline {section}{\numberline {50.35}LLL}{836}
\contentsline {section}{\numberline {50.36}OrthogonalEmbeddings}{836}
\contentsline {section}{\numberline {50.37}ShortestVectors}{838}
\contentsline {section}{\numberline {50.38}Extract}{838}
\contentsline {section}{\numberline {50.39}Decreased}{839}
\contentsline {section}{\numberline {50.40}DnLattice}{840}
\contentsline {section}{\numberline {50.41}ContainedDecomposables}{841}
\contentsline {section}{\numberline {50.42}ContainedCharacters}{842}
\contentsline {section}{\numberline {50.43}ContainedSpecialVectors}{842}
\contentsline {section}{\numberline {50.44}ContainedPossibleCharacters}{843}
\contentsline {section}{\numberline {50.45}ContainedPossibleVirtualCharacters}{843}
\contentsline {chapter}{\numberline {51}Maps and Parametrized Maps}{845}
\contentsline {section}{\numberline {51.1}More about Maps and Parametrized Maps}{845}
\contentsline {section}{\numberline {51.2}CompositionMaps}{846}
\contentsline {section}{\numberline {51.3}InverseMap}{846}
\contentsline {section}{\numberline {51.4}ProjectionMap}{847}
\contentsline {section}{\numberline {51.5}Parametrized}{847}
\contentsline {section}{\numberline {51.6}ContainedMaps}{847}
\contentsline {section}{\numberline {51.7}UpdateMap}{848}
\contentsline {section}{\numberline {51.8}CommutativeDiagram}{848}
\contentsline {section}{\numberline {51.9}TransferDiagram}{849}
\contentsline {section}{\numberline {51.10}Indeterminateness}{850}
\contentsline {section}{\numberline {51.11}PrintAmbiguity}{850}
\contentsline {section}{\numberline {51.12}Powermap}{851}
\contentsline {section}{\numberline {51.13}SubgroupFusions}{851}
\contentsline {section}{\numberline {51.14}InitPowermap}{852}
\contentsline {section}{\numberline {51.15}Congruences}{853}
\contentsline {section}{\numberline {51.16}ConsiderKernels}{854}
\contentsline {section}{\numberline {51.17}ConsiderSmallerPowermaps}{854}
\contentsline {section}{\numberline {51.18}InitFusion}{855}
\contentsline {section}{\numberline {51.19}CheckPermChar}{855}
\contentsline {section}{\numberline {51.20}CheckFixedPoints}{856}
\contentsline {section}{\numberline {51.21}TestConsistencyMaps}{856}
\contentsline {section}{\numberline {51.22}ConsiderTableAutomorphisms}{857}
\contentsline {section}{\numberline {51.23}PowermapsAllowedBySymmetrisations}{857}
\contentsline {section}{\numberline {51.24}FusionsAllowedByRestrictions}{858}
\contentsline {section}{\numberline {51.25}OrbitFusions}{859}
\contentsline {section}{\numberline {51.26}OrbitPowermaps}{860}
\contentsline {section}{\numberline {51.27}RepresentativesFusions}{860}
\contentsline {section}{\numberline {51.28}RepresentativesPowermaps}{861}
\contentsline {section}{\numberline {51.29}Indirected}{861}
\contentsline {section}{\numberline {51.30}Powmap}{862}
\contentsline {section}{\numberline {51.31}ElementOrdersPowermap}{862}
\contentsline {chapter}{\numberline {52}Character Table Libraries}{865}
\contentsline {section}{\numberline {52.1}Contents of the Table Libraries}{865}
\contentsline {section}{\numberline {52.2}Selecting Library Tables}{867}
\contentsline {section}{\numberline {52.3}ATLAS Tables}{868}
\contentsline {section}{\numberline {52.4}Examples of the ATLAS format for GAP tables}{871}
\contentsline {section}{\numberline {52.5}CAS Tables}{875}
\contentsline {section}{\numberline {52.6}Organization of the Table Libraries}{875}
\contentsline {section}{\numberline {52.7}How to Extend a Table Library}{877}
\contentsline {section}{\numberline {52.8}FirstNameCharTable}{878}
\contentsline {section}{\numberline {52.9}FileNameCharTable}{878}
\contentsline {chapter}{\numberline {53}Class Functions}{879}
\contentsline {section}{\numberline {53.1}Why Group Characters}{879}
\contentsline {section}{\numberline {53.2}More about Class Functions}{881}
\contentsline {section}{\numberline {53.3}Operators for Class Functions}{882}
\contentsline {section}{\numberline {53.4}Functions for Class Functions}{883}
\contentsline {section}{\numberline {53.5}ClassFunction}{884}
\contentsline {section}{\numberline {53.6}VirtualCharacter}{885}
\contentsline {section}{\numberline {53.7}Character}{885}
\contentsline {section}{\numberline {53.8}IsClassFunction}{886}
\contentsline {section}{\numberline {53.9}IsVirtualCharacter}{886}
\contentsline {section}{\numberline {53.10}IsCharacter}{886}
\contentsline {section}{\numberline {53.11}Irr}{887}
\contentsline {section}{\numberline {53.12}InertiaSubgroup}{887}
\contentsline {section}{\numberline {53.13}OrbitsCharacters}{887}
\contentsline {section}{\numberline {53.14}Storing Subgroup Information}{888}
\contentsline {section}{\numberline {53.15}NormalSubgroupClasses}{889}
\contentsline {section}{\numberline {53.16}ClassesNormalSubgroup}{890}
\contentsline {section}{\numberline {53.17}FactorGroupNormalSubgroupClasses}{890}
\contentsline {section}{\numberline {53.18}Class Function Records}{890}
\contentsline {chapter}{\numberline {54}Monomiality Questions}{891}
\contentsline {section}{\numberline {54.1}More about Monomiality Questions}{891}
\contentsline {section}{\numberline {54.2}Alpha}{892}
\contentsline {section}{\numberline {54.3}Delta}{893}
\contentsline {section}{\numberline {54.4}BergerCondition}{893}
\contentsline {section}{\numberline {54.5}TestHomogeneous}{893}
\contentsline {section}{\numberline {54.6}TestQuasiPrimitive}{894}
\contentsline {section}{\numberline {54.7}IsPrimitive for Characters}{895}
\contentsline {section}{\numberline {54.8}TestInducedFromNormalSubgroup}{895}
\contentsline {section}{\numberline {54.9}TestSubnormallyMonomial}{896}
\contentsline {section}{\numberline {54.10}TestMonomialQuick}{897}
\contentsline {section}{\numberline {54.11}TestMonomial}{897}
\contentsline {section}{\numberline {54.12}TestRelativelySM}{898}
\contentsline {section}{\numberline {54.13}IsMinimalNonmonomial}{899}
\contentsline {section}{\numberline {54.14}MinimalNonmonomialGroup}{899}
\contentsline {chapter}{\numberline {55}Getting and Installing GAP}{901}
\contentsline {section}{\numberline {55.1}Getting GAP}{901}
\contentsline {section}{\numberline {55.2}Upgrading GAP}{903}
\contentsline {section}{\numberline {55.3}GAP for UNIX}{907}
\contentsline {section}{\numberline {55.4}Installation of GAP for UNIX}{908}
\contentsline {section}{\numberline {55.5}Features of GAP for UNIX}{912}
\contentsline {section}{\numberline {55.6}GAP for MS-DOS}{914}
\contentsline {section}{\numberline {55.7}Copyright of GAP for MS-DOS}{915}
\contentsline {section}{\numberline {55.8}Installation of GAP for MS-DOS}{916}
\contentsline {section}{\numberline {55.9}Features of GAP for MS-DOS}{920}
\contentsline {section}{\numberline {55.10}GAP for TOS}{922}
\contentsline {section}{\numberline {55.11}Copyright of GAP for TOS}{923}
\contentsline {section}{\numberline {55.12}Installation of GAP for TOS}{923}
\contentsline {section}{\numberline {55.13}Features of GAP for TOS}{927}
\contentsline {section}{\numberline {55.14}Porting GAP}{930}
\contentsline {chapter}{\numberline {56}Share Libraries}{933}
\contentsline {section}{\numberline {56.1}RequirePackage}{934}
\contentsline {section}{\numberline {56.2}ANU pq Package}{935}
\contentsline {section}{\numberline {56.3}Installing the ANU pq Package}{935}
\contentsline {section}{\numberline {56.4}ANU Sq Package}{942}
\contentsline {section}{\numberline {56.5}Installing the ANU Sq Package}{943}
\contentsline {section}{\numberline {56.6}GRAPE Package}{946}
\contentsline {section}{\numberline {56.7}Installing the GRAPE Package}{947}
\contentsline {section}{\numberline {56.8}MeatAxe Package}{950}
\contentsline {section}{\numberline {56.9}Installing the MeatAxe Package}{950}
\contentsline {section}{\numberline {56.10}NQ Package}{952}
\contentsline {section}{\numberline {56.11}Installing the NQ Package}{952}
\contentsline {section}{\numberline {56.12}SISYPHOS Package}{954}
\contentsline {section}{\numberline {56.13}Installing the SISYPHOS Package}{955}
\contentsline {section}{\numberline {56.14}Smash Package}{957}
\contentsline {section}{\numberline {56.15}Installing the Smash Package}{957}
\contentsline {section}{\numberline {56.16}Vector Enumeration Package}{958}
\contentsline {section}{\numberline {56.17}Installing the Vector Enumeration Package}{958}
\contentsline {section}{\numberline {56.18}Weyl Package}{960}
\contentsline {section}{\numberline {56.19}Installing the Weyl Package}{960}
\contentsline {section}{\numberline {56.20}The XGap Package}{962}
\contentsline {section}{\numberline {56.21}Installing the XGap Package}{962}
\contentsline {chapter}{\numberline {57}ANU Pq}{965}
\contentsline {section}{\numberline {57.1}Pq}{965}
\contentsline {section}{\numberline {57.2}PqHomomorphism}{966}
\contentsline {section}{\numberline {57.3}PqDescendants}{966}
\contentsline {section}{\numberline {57.4}PqList}{969}
\contentsline {section}{\numberline {57.5}SavePqList}{970}
\contentsline {section}{\numberline {57.6}StandardPresentation}{970}
\contentsline {section}{\numberline {57.7}IsIsomorphicPGroup}{972}
\contentsline {chapter}{\numberline {58}Cohomology}{973}
\contentsline {section}{\numberline {58.1}CHR}{974}
\contentsline {section}{\numberline {58.2}SchurMultiplier}{974}
\contentsline {section}{\numberline {58.3}CoveringGroup}{974}
\contentsline {section}{\numberline {58.4}FirstCohomologyDimension}{974}
\contentsline {section}{\numberline {58.5}SecondCohomologyDimension}{974}
\contentsline {section}{\numberline {58.6}SplitExtension}{975}
\contentsline {section}{\numberline {58.7}NonsplitExtension}{975}
\contentsline {section}{\numberline {58.8}CalcPres}{975}
\contentsline {section}{\numberline {58.9}PermRep}{976}
\contentsline {section}{\numberline {58.10}Further Information}{976}
\contentsline {chapter}{\numberline {59}Grape}{979}
\contentsline {section}{\numberline {59.1}Functions to construct and modify graphs}{980}
\contentsline {section}{\numberline {59.2}Graph}{980}
\contentsline {section}{\numberline {59.3}EdgeOrbitsGraph}{981}
\contentsline {section}{\numberline {59.4}NullGraph}{982}
\contentsline {section}{\numberline {59.5}CompleteGraph}{982}
\contentsline {section}{\numberline {59.6}JohnsonGraph}{983}
\contentsline {section}{\numberline {59.7}AddEdgeOrbit}{983}
\contentsline {section}{\numberline {59.8}RemoveEdgeOrbit}{984}
\contentsline {section}{\numberline {59.9}AssignVertexNames}{984}
\contentsline {section}{\numberline {59.10}Functions to inspect graphs, vertices and edges}{985}
\contentsline {section}{\numberline {59.11}IsGraph}{985}
\contentsline {section}{\numberline {59.12}OrderGraph}{985}
\contentsline {section}{\numberline {59.13}IsVertex}{985}
\contentsline {section}{\numberline {59.14}VertexName}{986}
\contentsline {section}{\numberline {59.15}Vertices}{986}
\contentsline {section}{\numberline {59.16}VertexDegree}{986}
\contentsline {section}{\numberline {59.17}VertexDegrees}{986}
\contentsline {section}{\numberline {59.18}IsLoopy}{986}
\contentsline {section}{\numberline {59.19}IsSimpleGraph}{987}
\contentsline {section}{\numberline {59.20}Adjacency}{987}
\contentsline {section}{\numberline {59.21}IsEdge}{987}
\contentsline {section}{\numberline {59.22}DirectedEdges}{987}
\contentsline {section}{\numberline {59.23}UndirectedEdges}{988}
\contentsline {section}{\numberline {59.24}Distance}{988}
\contentsline {section}{\numberline {59.25}Diameter}{988}
\contentsline {section}{\numberline {59.26}Girth}{989}
\contentsline {section}{\numberline {59.27}IsConnectedGraph}{989}
\contentsline {section}{\numberline {59.28}IsBipartite}{989}
\contentsline {section}{\numberline {59.29}IsNullGraph}{990}
\contentsline {section}{\numberline {59.30}IsCompleteGraph}{990}
\contentsline {section}{\numberline {59.31}Functions to determine regularity properties of graphs}{990}
\contentsline {section}{\numberline {59.32}IsRegularGraph}{991}
\contentsline {section}{\numberline {59.33}LocalParameters}{991}
\contentsline {section}{\numberline {59.34}GlobalParameters}{991}
\contentsline {section}{\numberline {59.35}IsDistanceRegular}{991}
\contentsline {section}{\numberline {59.36}CollapsedAdjacencyMat}{992}
\contentsline {section}{\numberline {59.37}OrbitalGraphIntersectionMatrices}{992}
\contentsline {section}{\numberline {59.38}Some special vertex subsets of a graph}{992}
\contentsline {section}{\numberline {59.39}ConnectedComponent}{992}
\contentsline {section}{\numberline {59.40}ConnectedComponents}{993}
\contentsline {section}{\numberline {59.41}Bicomponents}{993}
\contentsline {section}{\numberline {59.42}DistanceSet}{993}
\contentsline {section}{\numberline {59.43}Layers}{993}
\contentsline {section}{\numberline {59.44}IndependentSet}{994}
\contentsline {section}{\numberline {59.45}Functions to construct new graphs from old}{994}
\contentsline {section}{\numberline {59.46}InducedSubgraph}{994}
\contentsline {section}{\numberline {59.47}DistanceSetInduced}{994}
\contentsline {section}{\numberline {59.48}DistanceGraph}{995}
\contentsline {section}{\numberline {59.49}ComplementGraph}{995}
\contentsline {section}{\numberline {59.50}PointGraph}{996}
\contentsline {section}{\numberline {59.51}EdgeGraph}{996}
\contentsline {section}{\numberline {59.52}UnderlyingGraph}{997}
\contentsline {section}{\numberline {59.53}QuotientGraph}{997}
\contentsline {section}{\numberline {59.54}BipartiteDouble}{998}
\contentsline {section}{\numberline {59.55}GeodesicsGraph}{998}
\contentsline {section}{\numberline {59.56}CollapsedIndependentOrbitsGraph}{999}
\contentsline {section}{\numberline {59.57}CollapsedCompleteOrbitsGraph}{999}
\contentsline {section}{\numberline {59.58}NewGroupGraph}{1000}
\contentsline {section}{\numberline {59.59}Vertex-Colouring and Complete Subgraphs}{1000}
\contentsline {section}{\numberline {59.60}VertexColouring}{1001}
\contentsline {section}{\numberline {59.61}CompleteSubgraphs}{1001}
\contentsline {section}{\numberline {59.62}CompleteSubgraphsOfGivenSize}{1001}
\contentsline {section}{\numberline {59.63}Functions depending on nauty}{1002}
\contentsline {section}{\numberline {59.64}AutGroupGraph}{1002}
\contentsline {section}{\numberline {59.65}IsIsomorphicGraph}{1002}
\contentsline {section}{\numberline {59.66}An example}{1003}
\contentsline {chapter}{\numberline {60}GUAVA}{1005}
\contentsline {section}{\numberline {60.1}Codewords}{1006}
\contentsline {section}{\numberline {60.2}Codeword}{1006}
\contentsline {section}{\numberline {60.3}IsCodeword}{1007}
\contentsline {section}{\numberline {60.4}Comparisons of Codewords}{1008}
\contentsline {section}{\numberline {60.5}Operations for Codewords}{1008}
\contentsline {section}{\numberline {60.6}VectorCodeword}{1009}
\contentsline {section}{\numberline {60.7}PolyCodeword}{1009}
\contentsline {section}{\numberline {60.8}TreatAsVector}{1009}
\contentsline {section}{\numberline {60.9}TreatAsPoly}{1010}
\contentsline {section}{\numberline {60.10}NullWord}{1010}
\contentsline {section}{\numberline {60.11}DistanceCodeword}{1011}
\contentsline {section}{\numberline {60.12}Support}{1011}
\contentsline {section}{\numberline {60.13}WeightCodeword}{1011}
\contentsline {section}{\numberline {60.14}Codes}{1012}
\contentsline {section}{\numberline {60.15}IsCode}{1014}
\contentsline {section}{\numberline {60.16}IsLinearCode}{1014}
\contentsline {section}{\numberline {60.17}IsCyclicCode}{1014}
\contentsline {section}{\numberline {60.18}Comparisons of Codes}{1015}
\contentsline {section}{\numberline {60.19}Operations for Codes}{1015}
\contentsline {section}{\numberline {60.20}Generating Unrestricted Codes}{1016}
\contentsline {section}{\numberline {60.21}ElementsCode}{1017}
\contentsline {section}{\numberline {60.22}HadamardCode}{1017}
\contentsline {section}{\numberline {60.23}ConferenceCode}{1018}
\contentsline {section}{\numberline {60.24}MOLSCode}{1018}
\contentsline {section}{\numberline {60.25}RandomCode}{1019}
\contentsline {section}{\numberline {60.26}NordstromRobinsonCode}{1019}
\contentsline {section}{\numberline {60.27}GreedyCode}{1019}
\contentsline {section}{\numberline {60.28}LexiCode}{1020}
\contentsline {section}{\numberline {60.29}Generating Linear Codes}{1020}
\contentsline {section}{\numberline {60.30}GeneratorMatCode}{1021}
\contentsline {section}{\numberline {60.31}CheckMatCode}{1021}
\contentsline {section}{\numberline {60.32}HammingCode}{1022}
\contentsline {section}{\numberline {60.33}ReedMullerCode}{1022}
\contentsline {section}{\numberline {60.34}ExtendedBinaryGolayCode}{1022}
\contentsline {section}{\numberline {60.35}ExtendedTernaryGolayCode}{1023}
\contentsline {section}{\numberline {60.36}AlternantCode}{1023}
\contentsline {section}{\numberline {60.37}GoppaCode}{1023}
\contentsline {section}{\numberline {60.38}GeneralizedSrivastavaCode}{1024}
\contentsline {section}{\numberline {60.39}SrivastavaCode}{1024}
\contentsline {section}{\numberline {60.40}CordaroWagnerCode}{1024}
\contentsline {section}{\numberline {60.41}RandomLinearCode}{1025}
\contentsline {section}{\numberline {60.42}BestKnownLinearCode}{1025}
\contentsline {section}{\numberline {60.43}Generating Cyclic Codes}{1026}
\contentsline {section}{\numberline {60.44}GeneratorPolCode}{1026}
\contentsline {section}{\numberline {60.45}CheckPolCode}{1027}
\contentsline {section}{\numberline {60.46}BinaryGolayCode}{1027}
\contentsline {section}{\numberline {60.47}TernaryGolayCode}{1027}
\contentsline {section}{\numberline {60.48}RootsCode}{1028}
\contentsline {section}{\numberline {60.49}BCHCode}{1028}
\contentsline {section}{\numberline {60.50}ReedSolomonCode}{1029}
\contentsline {section}{\numberline {60.51}QRCode}{1029}
\contentsline {section}{\numberline {60.52}FireCode}{1030}
\contentsline {section}{\numberline {60.53}WholeSpaceCode}{1030}
\contentsline {section}{\numberline {60.54}NullCode}{1030}
\contentsline {section}{\numberline {60.55}RepetitionCode}{1031}
\contentsline {section}{\numberline {60.56}CyclicCodes}{1031}
\contentsline {section}{\numberline {60.57}Manipulating Codes}{1031}
\contentsline {section}{\numberline {60.58}ExtendedCode}{1032}
\contentsline {section}{\numberline {60.59}PuncturedCode}{1032}
\contentsline {section}{\numberline {60.60}EvenWeightSubcode}{1033}
\contentsline {section}{\numberline {60.61}PermutedCode}{1033}
\contentsline {section}{\numberline {60.62}ExpurgatedCode}{1034}
\contentsline {section}{\numberline {60.63}AugmentedCode}{1034}
\contentsline {section}{\numberline {60.64}RemovedElementsCode}{1035}
\contentsline {section}{\numberline {60.65}AddedElementsCode}{1035}
\contentsline {section}{\numberline {60.66}ShortenedCode}{1036}
\contentsline {section}{\numberline {60.67}LengthenedCode}{1037}
\contentsline {section}{\numberline {60.68}ResidueCode}{1037}
\contentsline {section}{\numberline {60.69}ConstructionBCode}{1037}
\contentsline {section}{\numberline {60.70}DualCode}{1038}
\contentsline {section}{\numberline {60.71}ConversionFieldCode}{1038}
\contentsline {section}{\numberline {60.72}CosetCode}{1038}
\contentsline {section}{\numberline {60.73}ConstantWeightSubcode}{1039}
\contentsline {section}{\numberline {60.74}StandardFormCode}{1039}
\contentsline {section}{\numberline {60.75}DirectSumCode}{1040}
\contentsline {section}{\numberline {60.76}UUVCode}{1041}
\contentsline {section}{\numberline {60.77}DirectProductCode}{1041}
\contentsline {section}{\numberline {60.78}IntersectionCode}{1041}
\contentsline {section}{\numberline {60.79}UnionCode}{1042}
\contentsline {section}{\numberline {60.80}Basic Functions for Codes}{1042}
\contentsline {section}{\numberline {60.81}Domain Functions for Codes}{1043}
\contentsline {section}{\numberline {60.82}Printing and Saving Codes}{1044}
\contentsline {section}{\numberline {60.83}GeneratorMat}{1045}
\contentsline {section}{\numberline {60.84}CheckMat}{1045}
\contentsline {section}{\numberline {60.85}GeneratorPol}{1046}
\contentsline {section}{\numberline {60.86}CheckPol}{1046}
\contentsline {section}{\numberline {60.87}RootsOfCode}{1046}
\contentsline {section}{\numberline {60.88}WordLength}{1047}
\contentsline {section}{\numberline {60.89}Redundancy}{1047}
\contentsline {section}{\numberline {60.90}MinimumDistance}{1047}
\contentsline {section}{\numberline {60.91}CoveringRadius}{1048}
\contentsline {section}{\numberline {60.92}WeightDistribution}{1049}
\contentsline {section}{\numberline {60.93}InnerDistribution}{1049}
\contentsline {section}{\numberline {60.94}OuterDistribution}{1049}
\contentsline {section}{\numberline {60.95}DistancesDistribution}{1050}
\contentsline {section}{\numberline {60.96}IsPerfectCode}{1050}
\contentsline {section}{\numberline {60.97}IsMDSCode}{1050}
\contentsline {section}{\numberline {60.98}IsSelfDualCode}{1051}
\contentsline {section}{\numberline {60.99}IsSelfOrthogonalCode}{1051}
\contentsline {section}{\numberline {60.100}IsEquivalent}{1052}
\contentsline {section}{\numberline {60.101}CodeIsomorphism}{1052}
\contentsline {section}{\numberline {60.102}AutomorphismGroup}{1052}
\contentsline {section}{\numberline {60.103}Decode}{1053}
\contentsline {section}{\numberline {60.104}Syndrome}{1053}
\contentsline {section}{\numberline {60.105}SyndromeTable}{1054}
\contentsline {section}{\numberline {60.106}StandardArray}{1054}
\contentsline {section}{\numberline {60.107}Display}{1055}
\contentsline {section}{\numberline {60.108}CodewordNr}{1056}
\contentsline {section}{\numberline {60.109}Code Records}{1056}
\contentsline {section}{\numberline {60.110}Bounds on codes}{1058}
\contentsline {section}{\numberline {60.111}UpperBoundSingleton}{1059}
\contentsline {section}{\numberline {60.112}UpperBoundHamming}{1059}
\contentsline {section}{\numberline {60.113}UpperBoundJohnson}{1060}
\contentsline {section}{\numberline {60.114}UpperBoundPlotkin}{1060}
\contentsline {section}{\numberline {60.115}UpperBoundElias}{1060}
\contentsline {section}{\numberline {60.116}UpperBoundGriesmer}{1061}
\contentsline {section}{\numberline {60.117}UpperBound}{1061}
\contentsline {section}{\numberline {60.118}OptimalityCode}{1061}
\contentsline {section}{\numberline {60.119}OptimalityLinearCode}{1062}
\contentsline {section}{\numberline {60.120}LowerBoundMinimumDistance}{1062}
\contentsline {section}{\numberline {60.121}UpperBoundMinimumDistance}{1062}
\contentsline {section}{\numberline {60.122}BoundsMinimumDistance}{1063}
\contentsline {section}{\numberline {60.123}Matrices for Codes}{1063}
\contentsline {section}{\numberline {60.124}KrawtchoukMat}{1064}
\contentsline {section}{\numberline {60.125}GrayMat}{1064}
\contentsline {section}{\numberline {60.126}SylvesterMat}{1064}
\contentsline {section}{\numberline {60.127}HadamardMat}{1065}
\contentsline {section}{\numberline {60.128}MOLS}{1065}
\contentsline {section}{\numberline {60.129}PutStandardForm}{1066}
\contentsline {section}{\numberline {60.130}IsInStandardForm}{1067}
\contentsline {section}{\numberline {60.131}PermutedCols}{1067}
\contentsline {section}{\numberline {60.132}VerticalConversionFieldMat}{1067}
\contentsline {section}{\numberline {60.133}HorizontalConversionFieldMat}{1068}
\contentsline {section}{\numberline {60.134}IsLatinSquare}{1068}
\contentsline {section}{\numberline {60.135}AreMOLS}{1068}
\contentsline {section}{\numberline {60.136}Miscellaneous functions}{1069}
\contentsline {section}{\numberline {60.137}SphereContent}{1069}
\contentsline {section}{\numberline {60.138}Krawtchouk}{1069}
\contentsline {section}{\numberline {60.139}PrimitiveUnityRoot}{1069}
\contentsline {section}{\numberline {60.140}ReciprocalPolynomial}{1070}
\contentsline {section}{\numberline {60.141}CyclotomicCosets}{1070}
\contentsline {section}{\numberline {60.142}WeightHistogram}{1071}
\contentsline {chapter}{\numberline {61}The MeatAxe}{1073}
\contentsline {section}{\numberline {61.1}More about the MeatAxe in GAP}{1074}
\contentsline {section}{\numberline {61.2}GapObject}{1074}
\contentsline {section}{\numberline {61.3}Using the MeatAxe in GAP. An Example}{1075}
\contentsline {section}{\numberline {61.4}MeatAxe Matrices}{1077}
\contentsline {section}{\numberline {61.5}MeatAxeMat}{1077}
\contentsline {section}{\numberline {61.6}Operations for MeatAxe Matrices}{1078}
\contentsline {section}{\numberline {61.7}Functions for MeatAxe Matrices}{1079}
\contentsline {section}{\numberline {61.8}BrauerCharacterValue}{1080}
\contentsline {section}{\numberline {61.9}MeatAxe Permutations}{1080}
\contentsline {section}{\numberline {61.10}MeatAxePerm}{1080}
\contentsline {section}{\numberline {61.11}Operations for MeatAxe Permutations}{1081}
\contentsline {section}{\numberline {61.12}Functions for MeatAxe Permutations}{1081}
\contentsline {section}{\numberline {61.13}MeatAxe Matrix Groups}{1081}
\contentsline {section}{\numberline {61.14}Functions for MeatAxe Matrix Groups}{1081}
\contentsline {section}{\numberline {61.15}MeatAxe Matrix Algebras}{1082}
\contentsline {section}{\numberline {61.16}Functions for MeatAxe Matrix Algebras}{1082}
\contentsline {section}{\numberline {61.17}MeatAxe Modules}{1083}
\contentsline {section}{\numberline {61.18}Set Theoretic Functions for MeatAxe Modules}{1083}
\contentsline {section}{\numberline {61.19}Vector Space Functions for MeatAxe Modules}{1083}
\contentsline {section}{\numberline {61.20}Module Functions for MeatAxe Modules}{1083}
\contentsline {section}{\numberline {61.21}MeatAxe.Unbind}{1085}
\contentsline {section}{\numberline {61.22}MeatAxe Object Records}{1085}
\contentsline {chapter}{\numberline {62}Sisyphos}{1089}
\contentsline {section}{\numberline {62.1}PrintSISYPHOSWord}{1089}
\contentsline {section}{\numberline {62.2}PrintSisyphosInputPGroup}{1090}
\contentsline {section}{\numberline {62.3}IsCompatiblePCentralSeries}{1091}
\contentsline {section}{\numberline {62.4}Automorphisms}{1091}
\contentsline {section}{\numberline {62.5}AgNormalizedAutomorphisms}{1092}
\contentsline {section}{\numberline {62.6}AgNormalizedOuterAutomorphisms}{1092}
\contentsline {section}{\numberline {62.7}IsIsomorphic}{1092}
\contentsline {section}{\numberline {62.8}Isomorphisms}{1093}
\contentsline {section}{\numberline {62.9}CorrespondingAutomorphism}{1094}
\contentsline {section}{\numberline {62.10}AutomorphismGroupElements}{1094}
\contentsline {section}{\numberline {62.11}NormalizedUnitsGroupRing}{1094}
\contentsline {chapter}{\numberline {63}Smash, Matrix Groups and $G$-Modules}{1097}
\contentsline {section}{\numberline {63.1}GModule}{1097}
\contentsline {section}{\numberline {63.2}IsGModule}{1097}
\contentsline {section}{\numberline {63.3}DualGMod}{1098}
\contentsline {section}{\numberline {63.4}InducedGMod}{1098}
\contentsline {section}{\numberline {63.5}PermGMod}{1098}
\contentsline {section}{\numberline {63.6}TensorProductGMod}{1098}
\contentsline {section}{\numberline {63.7}WedgeGMod}{1098}
\contentsline {section}{\numberline {63.8}WreathProd}{1098}
\contentsline {section}{\numberline {63.9}WreathPower}{1099}
\contentsline {section}{\numberline {63.10}IsIrredGMod}{1099}
\contentsline {section}{\numberline {63.11}SubQuotGMod}{1099}
\contentsline {section}{\numberline {63.12}SpinBasis}{1100}
\contentsline {section}{\numberline {63.13}IsAbsIrredGMod}{1100}
\contentsline {section}{\numberline {63.14}FieldGenCentMat}{1100}
\contentsline {section}{\numberline {63.15}IsomGMod}{1100}
\contentsline {section}{\numberline {63.16}HomGMod}{1101}
\contentsline {section}{\numberline {63.17}MinSubGMods}{1101}
\contentsline {section}{\numberline {63.18}ChopGMod}{1101}
\contentsline {section}{\numberline {63.19}SmashGMod}{1101}
\contentsline {section}{\numberline {63.20}SemiLinearDecomp}{1102}
\contentsline {section}{\numberline {63.21}TensorProductDecomp}{1102}
\contentsline {section}{\numberline {63.22}SymTensorProductDecomp}{1103}
\contentsline {section}{\numberline {63.23}ExtraSpecialDecomp}{1103}
\contentsline {section}{\numberline {63.24}IsPrimitiveGMod}{1104}
\contentsline {section}{\numberline {63.25}MinBlocks}{1104}
\contentsline {section}{\numberline {63.26}BlockSystemFlag}{1104}
\contentsline {section}{\numberline {63.27}InitialiseSeed}{1104}
\contentsline {section}{\numberline {63.28}RandomElement}{1105}
\contentsline {section}{\numberline {63.29}ChooseRandomElements}{1105}
\contentsline {section}{\numberline {63.30}ElementOfOrder}{1105}
\contentsline {section}{\numberline {63.31}ElementWithCharPol}{1105}
\contentsline {section}{\numberline {63.32}LargestPrimeOrderElement}{1105}
\contentsline {section}{\numberline {63.33}LargestPrimePowerOrderElement}{1106}
\contentsline {section}{\numberline {63.34}MatrixOrder}{1106}
\contentsline {section}{\numberline {63.35}Other matrix functions}{1106}
\contentsline {section}{\numberline {63.36}Components of a $G$-module record}{1106}
\contentsline {section}{\numberline {63.37}Examples}{1108}
\contentsline {chapter}{\numberline {64}Vector Enumeration}{1115}
\contentsline {section}{\numberline {64.1}Operation for Finitely Presented Algebras}{1115}
\contentsline {section}{\numberline {64.2}More about Vector Enumeration}{1116}
\contentsline {section}{\numberline {64.3}Examples of Vector Enumeration}{1118}
\contentsline {section}{\numberline {64.4}Using Vector Enumeration with the MeatAxe}{1121}
\contentsline {chapter}{\numberline {65}Weyl Groups and Hecke Algebras}{1123}
\contentsline {section}{\numberline {65.1}CartanMat}{1131}
\contentsline {section}{\numberline {65.2}DirectSumCartanMat}{1131}
\contentsline {section}{\numberline {65.3}SimpleReflectionMatrices}{1132}
\contentsline {section}{\numberline {65.4}Rootsystem}{1132}
\contentsline {section}{\numberline {65.5}PermRepresentationRoots}{1132}
\contentsline {section}{\numberline {65.6}Weyl}{1133}
\contentsline {section}{\numberline {65.7}PermWeylWord}{1134}
\contentsline {section}{\numberline {65.8}WeylWordPerm}{1134}
\contentsline {section}{\numberline {65.9}WeylLengthPerm}{1134}
\contentsline {section}{\numberline {65.10}ReducedWeylWord}{1135}
\contentsline {section}{\numberline {65.11}LongestWeylWord}{1135}
\contentsline {section}{\numberline {65.12}WeylReflections}{1135}
\contentsline {section}{\numberline {65.13}WeylRightCosetRepresentatives}{1135}
\contentsline {section}{\numberline {65.14}WeylCosetPermRepresentation}{1136}
\contentsline {section}{\numberline {65.15}WeylElements}{1136}
\contentsline {section}{\numberline {65.16}WeylConjugacyClasses}{1136}
\contentsline {section}{\numberline {65.17}ParametersCentralizers}{1137}
\contentsline {section}{\numberline {65.18}Bruhat}{1137}
\contentsline {section}{\numberline {65.19}KazhdanLusztigPolynomial}{1138}
\contentsline {section}{\numberline {65.20}KLCoefficient}{1138}
\contentsline {section}{\numberline {65.21}WeylMueMat}{1138}
\contentsline {section}{\numberline {65.22}DecomposedLeftCells}{1139}
\contentsline {section}{\numberline {65.23}LeftCells}{1139}
\contentsline {section}{\numberline {65.24}LeftCellRepresentation}{1140}
\contentsline {section}{\numberline {65.25}Hecke}{1140}
\contentsline {section}{\numberline {65.26}WeylClassPolynomials}{1140}
\contentsline {section}{\numberline {65.27}HeckeReflectionRepresentation}{1141}
\contentsline {section}{\numberline {65.28}CheckHeckeDefiningRelations}{1141}
\contentsline {section}{\numberline {65.29}CharHeckeRepresentation}{1141}
\contentsline {section}{\numberline {65.30}HeckeCharTable}{1142}
