GAP - Reference Manual Release 4.8.10, 15-Jan-2018 The GAP Group The GAP Group Email: mailto:support@gap-system.org Homepage: https://www.gap-system.org ------------------------------------------------------- Copyright Copyright © (1987-2018) for the core part of the GAP system by the GAP Group. Most parts of this distribution, including the core part of the GAP system are distributed under the terms of the GNU General Public License, see http://www.gnu.org/licenses/gpl.html or the file GPL in the etc directory of the GAP installation. More detailed information about copyright and licenses of parts of this distribution can be found in Section 1.4 of this manual. GAP is developed over a long time and has many authors and contributors. More detailed information can be found in Section 1.2 of this manual. ------------------------------------------------------- Contents (ref) 1 Preface 1.1 The GAP System 1.2 Authors and Maintainers 1.3 Acknowledgements 1.4 Copyright and License 1.5 Further Information about GAP 2 The Help System 2.1 Invoking the Help 2.2 Browsing through the Sections 2.3 Changing the Help Viewer 2.3-1 SetHelpViewer 2.4 The Pager Command 2.4-1 Pager 3 Running GAP 3.1 Command Line Options 3.2 The gap.ini and gaprc files 3.2-1 The gap.ini file 3.2-2 The gaprc file 3.2-3 Configuring User preferences 3.2-4 DeclareUserPreference 3.3 Saving and Loading a Workspace 3.3-1 SaveWorkspace 3.4 Testing for the System Architecture 3.4-1 ARCH_IS_UNIX 3.4-2 ARCH_IS_MAC_OS_X 3.4-3 ARCH_IS_WINDOWS 3.5 Global Values that Control the GAP Session 3.5-1 GAPInfo 3.6 Coloring the Prompt and Input 3.6-1 ColorPrompt 4 The Programming Language 4.1 Language Overview 4.2 Lexical Structure 4.3 Symbols 4.4 Whitespaces 4.5 Keywords 4.6 Identifiers 4.6-1 IsValidIdentifier 4.7 Expressions 4.8 Variables 4.8-1 IsBound 4.8-2 Unbind 4.9 More About Global Variables 4.9-1 IsReadOnlyGlobal 4.9-2 MakeReadOnlyGlobal 4.9-3 MakeReadWriteGlobal 4.9-4 ValueGlobal 4.9-5 IsBoundGlobal 4.9-6 UnbindGlobal 4.9-7 BindGlobal 4.9-8 NamesGVars 4.9-9 NamesSystemGVars 4.9-10 NamesUserGVars 4.9-11 TemporaryGlobalVarName 4.10 Namespaces for GAP packages 4.11 Function Calls 4.11-1 Function Call With Arguments 4.11-2 Function Call With Options 4.12 Comparisons 4.13 Arithmetic Operators 4.14 Statements 4.15 Assignments 4.16 Procedure Calls 4.17 If 4.18 While 4.19 Repeat 4.20 For 4.21 Break 4.22 Continue 4.23 Function 4.24 Return (With or without Value) 5 Functions 5.1 Information about a function 5.1-1 NameFunction 5.1-2 NumberArgumentsFunction 5.1-3 NamesLocalVariablesFunction 5.1-4 FilenameFunc 5.1-5 StartlineFunc 5.1-6 PageSource 5.2 Calling a function with a list argument that is interpreted as several arguments 5.2-1 CallFuncList 5.3 Calling a function with a time limit 5.3-1 CallWithTimeout 5.3-2 GAPInfo.TimeoutsSupported 5.4 Functions that do nothing 5.4-1 ReturnTrue 5.4-2 ReturnFalse 5.4-3 ReturnFail 5.4-4 ReturnNothing 5.4-5 ReturnFirst 5.4-6 IdFunc 5.5 Function Types 5.5-1 IsFunction 5.5-2 IsOperation 5.5-3 FunctionsFamily 5.6 Naming Conventions 6 Main Loop and Break Loop 6.1 Main Loop 6.2 Special Rules for Input Lines 6.3 View and Print 6.3-1 Default delegations in the library 6.3-2 Recommendations for the implementation 6.3-3 View 6.3-4 Print 6.3-5 ViewObj 6.3-6 Display 6.3-7 SetNameObject 6.4 Break Loops 6.4-1 quit from a break loop 6.4-2 return from a break loop 6.4-3 OnBreak 6.4-4 OnBreakMessage 6.4-5 Where 6.5 Variable Access in a Break Loop 6.5-1 DownEnv and UpEnv 6.6 Error and ErrorCount 6.6-1 Error 6.6-2 ErrorNoReturn 6.6-3 ErrorCount 6.7 Leaving GAP 6.7-1 QUIT 6.7-2 GAP_EXIT_CODE 6.7-3 QUIT_GAP 6.7-4 FORCE_QUIT_GAP 6.7-5 InstallAtExit 6.7-6 SaveOnExitFile 6.8 Line Editing 6.9 Editing using the readline library 6.9-1 Readline customization 6.9-2 The command line history 6.9-3 SaveCommandLineHistory 6.9-4 Writing your own command line editing functions 6.10 Editing Files 6.10-1 Edit 6.11 Editor Support 6.12 Changing the Screen Size 6.12-1 SizeScreen 6.13 Teaching Mode 6.13-1 TeachingMode 7 Debugging and Profiling Facilities 7.1 Recovery from NoMethodFound-Errors 7.1-1 ShowArguments 7.1-2 ShowArgument 7.1-3 ShowDetails 7.1-4 ShowMethods 7.1-5 ShowOtherMethods 7.2 Inspecting Applicable Methods 7.2-1 ApplicableMethod 7.3 Tracing Methods 7.3-1 TraceMethods 7.3-2 TraceAllMethods 7.3-3 UntraceMethods 7.3-4 UntraceAllMethods 7.3-5 TraceImmediateMethods 7.4 Info Functions 7.4-1 NewInfoClass 7.4-2 DeclareInfoClass 7.4-3 SetInfoLevel 7.4-4 InfoLevel 7.4-5 Info 7.4-6 Customizing Info (7.4-5) statements 7.4-7 InfoWarning 7.5 Assertions 7.5-1 SetAssertionLevel 7.5-2 AssertionLevel 7.5-3 Assert 7.6 Timing 7.6-1 Runtimes 7.6-2 Runtime 7.6-3 time 7.7 Profiling 7.7-1 Function Profiling 7.7-2 ProfileGlobalFunctions 7.7-3 ProfileOperations 7.7-4 ProfileOperationsAndMethods 7.7-5 ProfileFunctions 7.7-6 UnprofileFunctions 7.7-7 ProfileMethods 7.7-8 UnprofileMethods 7.7-9 DisplayProfile 7.7-10 ClearProfile 7.7-11 An Example of Function Profiling 7.7-12 Line By Line Profiling 7.7-13 Line by Line profiling example 7.7-14 ProfileLineByLine 7.7-15 CoverageLineByLine 7.7-16 UnprofileLineByLine 7.7-17 UncoverageLineByLine 7.7-18 ActivateProfileColour 7.7-19 IsLineByLineProfileActive 7.7-20 DisplayCacheStats 7.7-21 ClearCacheStats 7.8 Information about the version used 7.9 Test Files 7.9-1 Starting and stopping test 7.9-2 Test 7.9-3 TestDirectory 7.10 Debugging Recursion 7.10-1 SetRecursionTrapInterval 7.11 Global Memory Information 7.11-1 GasmanStatistics 7.11-2 GasmanMessageStatus 7.11-3 GasmanLimits 8 Options Stack 8.1 Functions Dealing with the Options Stack 8.1-1 PushOptions 8.1-2 PopOptions 8.1-3 ResetOptionsStack 8.1-4 OnQuit 8.1-5 ValueOption 8.1-6 DisplayOptionsStack 8.1-7 InfoOptions 8.2 Options Stack – an Example 9 Files and Filenames 9.1 Portability 9.1-1 LastSystemError 9.2 GAP Root Directories 9.3 Directories 9.3-1 IsDirectory 9.3-2 Directory 9.3-3 DirectoryTemporary 9.3-4 DirectoryCurrent 9.3-5 DirectoriesLibrary 9.3-6 DirectoriesSystemPrograms 9.3-7 DirectoryContents 9.3-8 DirectoryDesktop 9.3-9 DirectoryHome 9.4 File Names 9.4-1 Filename 9.5 Special Filenames 9.6 File Access 9.6-1 IsExistingFile 9.6-2 IsReadableFile 9.6-3 IsWritableFile 9.6-4 IsExecutableFile 9.6-5 IsDirectoryPath 9.7 File Operations 9.7-1 Read 9.7-2 ReadAsFunction 9.7-3 PrintTo and AppendTo 9.7-4 LogTo 9.7-5 InputLogTo 9.7-6 OutputLogTo 9.7-7 CrcFile 9.7-8 RemoveFile 9.7-9 Reread 10 Streams 10.1 Categories for Streams and the StreamsFamily 10.1-1 IsStream 10.1-2 IsClosedStream 10.1-3 IsInputStream 10.1-4 IsInputTextStream 10.1-5 IsInputTextNone 10.1-6 IsOutputStream 10.1-7 IsOutputTextStream 10.1-8 IsOutputTextNone 10.1-9 StreamsFamily 10.2 Operations applicable to All Streams 10.2-1 CloseStream 10.2-2 FileDescriptorOfStream 10.2-3 UNIXSelect 10.3 Operations for Input Streams 10.3-1 Read 10.3-2 ReadAsFunction 10.3-3 ReadByte 10.3-4 ReadLine 10.3-5 ReadAll 10.3-6 IsEndOfStream 10.3-7 PositionStream 10.3-8 RewindStream 10.3-9 SeekPositionStream 10.4 Operations for Output Streams 10.4-1 WriteByte 10.4-2 WriteLine 10.4-3 WriteAll 10.4-4 PrintTo and AppendTo (for streams) 10.4-5 LogTo 10.4-6 InputLogTo 10.4-7 OutputLogTo 10.4-8 SetPrintFormattingStatus 10.5 File Streams 10.5-1 InputTextFile 10.5-2 OutputTextFile 10.6 User Streams 10.6-1 InputTextUser 10.6-2 OutputTextUser 10.6-3 InputFromUser 10.7 String Streams 10.7-1 InputTextString 10.7-2 OutputTextString 10.8 Input-Output Streams 10.8-1 IsInputOutputStream 10.8-2 InputOutputLocalProcess 10.8-3 ReadAllLine 10.9 Dummy Streams 10.9-1 InputTextNone 10.9-2 OutputTextNone 10.10 Handling of Streams in the Background 10.10-1 InstallCharReadHookFunc 10.10-2 UnInstallCharReadHookFunc 10.11 Comma separated files 10.11-1 ReadCSV 10.11-2 PrintCSV 11 Processes 11.1 Process and Exec 11.1-1 Process 11.1-2 Exec 12 Objects and Elements 12.1 Objects 12.1-1 IsObject 12.2 Elements as equivalence classes 12.3 Sets 12.4 Domains 12.5 Identical Objects 12.5-1 IsIdenticalObj 12.5-2 IsNotIdenticalObj 12.6 Mutability and Copyability 12.6-1 IsCopyable 12.6-2 IsMutable 12.6-3 Immutable 12.6-4 MakeImmutable 12.6-5 Mutability of Iterators 12.6-6 Mutability of Results of Arithmetic Operations 12.7 Duplication of Objects 12.7-1 ShallowCopy 12.7-2 StructuralCopy 12.8 Other Operations Applicable to any Object 12.8-1 SetName 12.8-2 Name 12.8-3 InfoText 12.8-4 IsInternallyConsistent 12.8-5 MemoryUsage 13 Types of Objects 13.1 Families 13.1-1 FamilyObj 13.2 Filters 13.2-1 RankFilter 13.2-2 NamesFilter 13.2-3 ShowImpliedFilters 13.3 Categories 13.3-1 CategoriesOfObject 13.4 Representation 13.4-1 RepresentationsOfObject 13.5 Attributes 13.5-1 KnownAttributesOfObject 13.6 Setter and Tester for Attributes 13.6-1 Tester 13.6-2 Setter 13.6-3 AttributeValueNotSet 13.6-4 InfoAttributes 13.6-5 DisableAttributeValueStoring 13.6-6 EnableAttributeValueStoring 13.7 Properties 13.7-1 KnownPropertiesOfObject 13.7-2 KnownTruePropertiesOfObject 13.8 Other Filters 13.9 Types 13.9-1 TypeObj 13.9-2 DataType 14 Integers 14.1 Integers: Global Variables 14.1-1 Integers 14.1-2 IsIntegers 14.2 Elementary Operations for Integers 14.2-1 IsInt 14.2-2 IsPosInt 14.2-3 Int 14.2-4 IsEvenInt 14.2-5 IsOddInt 14.2-6 AbsInt 14.2-7 SignInt 14.2-8 LogInt 14.2-9 RootInt 14.2-10 SmallestRootInt 14.2-11 ListOfDigits 14.2-12 Random 14.3 Quotients and Remainders 14.3-1 QuoInt 14.3-2 BestQuoInt 14.3-3 RemInt 14.3-4 GcdInt 14.3-5 Gcdex 14.3-6 LcmInt 14.3-7 CoefficientsQadic 14.3-8 CoefficientsMultiadic 14.3-9 ChineseRem 14.3-10 PowerModInt 14.4 Prime Integers and Factorization 14.4-1 Primes 14.4-2 IsPrimeInt 14.4-3 PrimalityProof 14.4-4 IsPrimePowerInt 14.4-5 NextPrimeInt 14.4-6 PrevPrimeInt 14.4-7 FactorsInt 14.4-8 PrimeDivisors 14.4-9 PartialFactorization 14.4-10 PrintFactorsInt 14.4-11 PrimePowersInt 14.4-12 DivisorsInt 14.5 Residue Class Rings 14.5-1 \mod 14.5-2 ZmodnZ 14.5-3 ZmodnZObj 14.5-4 IsZmodnZObj 14.6 Check Digits 14.6-1 CheckDigitISBN 14.6-2 CheckDigitTestFunction 14.7 Random Sources 14.7-1 IsRandomSource 14.7-2 Random 14.7-3 State 14.7-4 IsMersenneTwister 14.7-5 RandomSource 15 Number Theory 15.1 InfoNumtheor (Info Class) 15.1-1 InfoNumtheor 15.2 Prime Residues 15.2-1 PrimeResidues 15.2-2 Phi 15.2-3 Lambda 15.2-4 GeneratorsPrimeResidues 15.3 Primitive Roots and Discrete Logarithms 15.3-1 OrderMod 15.3-2 LogMod 15.3-3 PrimitiveRootMod 15.3-4 IsPrimitiveRootMod 15.4 Roots Modulo Integers 15.4-1 Jacobi 15.4-2 Legendre 15.4-3 RootMod 15.4-4 RootsMod 15.4-5 RootsUnityMod 15.5 Multiplicative Arithmetic Functions 15.5-1 Sigma 15.5-2 Tau 15.5-3 MoebiusMu 15.6 Continued Fractions 15.6-1 ContinuedFractionExpansionOfRoot 15.6-2 ContinuedFractionApproximationOfRoot 15.7 Miscellaneous 15.7-1 TwoSquares 16 Combinatorics 16.1 Combinatorial Numbers 16.1-1 Factorial 16.1-2 Binomial 16.1-3 Bell 16.1-4 Bernoulli 16.1-5 Stirling1 16.1-6 Stirling2 16.2 Combinations, Arrangements and Tuples 16.2-1 Combinations 16.2-2 Iterator and enumerator of combinations 16.2-3 NrCombinations 16.2-4 Arrangements 16.2-5 NrArrangements 16.2-6 UnorderedTuples 16.2-7 NrUnorderedTuples 16.2-8 Tuples 16.2-9 EnumeratorOfTuples 16.2-10 IteratorOfTuples 16.2-11 NrTuples 16.2-12 PermutationsList 16.2-13 NrPermutationsList 16.2-14 Derangements 16.2-15 NrDerangements 16.2-16 PartitionsSet 16.2-17 NrPartitionsSet 16.2-18 Partitions 16.2-19 IteratorOfPartitions 16.2-20 NrPartitions 16.2-21 OrderedPartitions 16.2-22 NrOrderedPartitions 16.2-23 PartitionsGreatestLE 16.2-24 PartitionsGreatestEQ 16.2-25 RestrictedPartitions 16.2-26 NrRestrictedPartitions 16.2-27 SignPartition 16.2-28 AssociatedPartition 16.2-29 PowerPartition 16.2-30 PartitionTuples 16.2-31 NrPartitionTuples 16.3 Fibonacci and Lucas Sequences 16.3-1 Fibonacci 16.3-2 Lucas 16.4 Permanent of a Matrix 16.4-1 Permanent 17 Rational Numbers 17.1 Rationals: Global Variables 17.1-1 Rationals 17.2 Elementary Operations for Rationals 17.2-1 IsRat 17.2-2 IsPosRat 17.2-3 IsNegRat 17.2-4 NumeratorRat 17.2-5 DenominatorRat 17.2-6 Rat 17.2-7 Random 18 Cyclotomic Numbers 18.1 Operations for Cyclotomics 18.1-1 E 18.1-2 Cyclotomics 18.1-3 IsCyclotomic 18.1-4 IsIntegralCyclotomic 18.1-5 Int 18.1-6 String 18.1-7 Conductor 18.1-8 AbsoluteValue 18.1-9 RoundCyc 18.1-10 CoeffsCyc 18.1-11 DenominatorCyc 18.1-12 ExtRepOfObj 18.1-13 DescriptionOfRootOfUnity 18.1-14 IsGaussInt 18.1-15 IsGaussRat 18.1-16 DefaultField 18.2 Infinity and negative Infinity 18.2-1 IsInfinity 18.3 Comparisons of Cyclotomics 18.4 ATLAS Irrationalities 18.4-1 EB, EC, ..., EH 18.4-2 EI and ER 18.4-3 EY, EX, ..., ES 18.4-4 EM, EL, ..., EJ 18.4-5 NK 18.4-6 AtlasIrrationality 18.5 Galois Conjugacy of Cyclotomics 18.5-1 GaloisCyc 18.5-2 ComplexConjugate 18.5-3 StarCyc 18.5-4 Quadratic 18.5-5 GaloisMat 18.5-6 RationalizedMat 18.6 Internally Represented Cyclotomics 18.6-1 SetCyclotomicsLimit 19 Floats 19.1 A sample run 19.2 Methods 19.2-1 Mathematical operations 19.2-2 EqFloat 19.2-3 PrecisionFloat 19.2-4 Interval operations 19.2-5 IsPInfinity 19.2-6 FLOAT 19.2-7 Float 19.2-8 Rat 19.2-9 SetFloats 19.3 High-precision-specific methods 19.4 Complex arithmetic 19.5 Interval-specific methods 20 Booleans 20.1 IsBool (Filter) 20.1-1 IsBool 20.2 Fail (Variable) 20.2-1 fail 20.3 Comparisons of Booleans 20.3-1 Equality and inequality of Booleans 20.3-2 Ordering of Booleans 20.4 Operations for Booleans 20.4-1 Logical disjunction 20.4-2 Logical conjunction 20.4-3 Logical negation 21 Lists 21.1 List Categories 21.1-1 IsList 21.1-2 IsDenseList 21.1-3 IsHomogeneousList 21.1-4 IsTable 21.1-5 IsRectangularTable 21.1-6 IsConstantTimeAccessList 21.2 Basic Operations for Lists 21.2-1 \[\] 21.3 List Elements 21.3-1 \{\} 21.4 List Assignment 21.4-1 \{\}\:\= 21.4-2 Add 21.4-3 Remove 21.4-4 CopyListEntries 21.4-5 Append 21.5 IsBound and Unbind for Lists 21.5-1 IsBound 21.5-2 Unbind 21.6 Identical Lists 21.7 Duplication of Lists 21.8 Membership Test for Lists 21.8-1 \in 21.9 Enlarging Internally Represented Lists 21.9-1 EmptyPlist 21.10 Comparisons of Lists 21.11 Arithmetic for Lists 21.12 Filters Controlling the Arithmetic Behaviour of Lists 21.12-1 IsGeneralizedRowVector 21.12-2 IsMultiplicativeGeneralizedRowVector 21.12-3 IsListDefault 21.12-4 NestingDepthA 21.12-5 NestingDepthM 21.13 Additive Arithmetic for Lists 21.13-1 Zero for lists 21.13-2 AdditiveInverse for lists 21.13-3 Addition of lists 21.13-4 Subtraction of lists 21.14 Multiplicative Arithmetic for Lists 21.14-1 One for lists 21.14-2 Inverse for lists 21.14-3 Multiplication of lists 21.14-4 Division of lists 21.14-5 mod for lists 21.14-6 Left quotients of lists 21.15 Mutability Status and List Arithmetic 21.15-1 ListWithIdenticalEntries 21.16 Finding Positions in Lists 21.16-1 Position 21.16-2 Positions 21.16-3 PositionCanonical 21.16-4 PositionNthOccurrence 21.16-5 PositionSorted 21.16-6 PositionSet 21.16-7 PositionProperty 21.16-8 PositionsProperty 21.16-9 PositionBound 21.16-10 PositionNot 21.16-11 PositionNonZero 21.16-12 PositionSublist 21.17 Properties and Attributes for Lists 21.17-1 IsMatchingSublist 21.17-2 IsDuplicateFree 21.17-3 IsSortedList 21.17-4 IsSSortedList 21.17-5 Length 21.17-6 ConstantTimeAccessList 21.18 Sorting Lists 21.18-1 Sort 21.18-2 SortParallel 21.18-3 Sortex 21.18-4 SortingPerm 21.19 Sorted Lists and Sets 21.19-1 \in 21.19-2 IsEqualSet 21.19-3 IsSubsetSet 21.19-4 AddSet 21.19-5 RemoveSet 21.19-6 UniteSet 21.19-7 IntersectSet 21.19-8 SubtractSet 21.20 Operations for Lists 21.20-1 Concatenation 21.20-2 Compacted 21.20-3 Collected 21.20-4 DuplicateFreeList 21.20-5 AsDuplicateFreeList 21.20-6 Flat 21.20-7 Reversed 21.20-8 Shuffle 21.20-9 IsLexicographicallyLess 21.20-10 Apply 21.20-11 Perform 21.20-12 PermListList 21.20-13 Maximum 21.20-14 Minimum 21.20-15 MaximumList and MinimumList 21.20-16 Cartesian 21.20-17 IteratorOfCartesianProduct 21.20-18 Permuted 21.20-19 List 21.20-20 Filtered 21.20-21 Number 21.20-22 First 21.20-23 ForAll 21.20-24 ForAny 21.20-25 Product 21.20-26 Sum 21.20-27 Iterated 21.20-28 ListN 21.21 Advanced List Manipulations 21.21-1 ListX 21.21-2 SetX 21.21-3 SumX 21.21-4 ProductX 21.22 Ranges 21.22-1 IsRange 21.22-2 ConvertToRangeRep 21.23 Enumerators 21.23-1 IsQuickPositionList 22 Boolean Lists 22.1 IsBlist (Filter) 22.1-1 IsBlist 22.2 Boolean Lists Representing Subsets 22.2-1 BlistList 22.2-2 ListBlist 22.2-3 SizeBlist 22.2-4 IsSubsetBlist 22.3 Set Operations via Boolean Lists 22.3-1 UnionBlist 22.3-2 IntersectionBlist 22.3-3 DifferenceBlist 22.4 Function that Modify Boolean Lists 22.4-1 UniteBlist 22.4-2 UniteBlistList 22.4-3 IntersectBlist 22.4-4 SubtractBlist 22.5 More about Boolean Lists 22.5-1 IsBlistRep 23 Row Vectors 23.1 IsRowVector (Filter) 23.1-1 IsRowVector 23.2 Operators for Row Vectors 23.2-1 NormedRowVector 23.3 Row Vectors over Finite Fields 23.3-1 ConvertToVectorRep 23.3-2 NumberFFVector 23.4 Coefficient List Arithmetic 23.4-1 AddRowVector 23.4-2 AddCoeffs 23.4-3 MultRowVector 23.4-4 CoeffsMod 23.5 Shifting and Trimming Coefficient Lists 23.5-1 LeftShiftRowVector 23.5-2 RightShiftRowVector 23.5-3 ShrinkRowVector 23.5-4 RemoveOuterCoeffs 23.6 Functions for Coding Theory 23.6-1 WeightVecFFE 23.6-2 DistanceVecFFE 23.6-3 DistancesDistributionVecFFEsVecFFE 23.6-4 DistancesDistributionMatFFEVecFFE 23.6-5 AClosestVectorCombinationsMatFFEVecFFE 23.6-6 CosetLeadersMatFFE 23.7 Vectors as coefficients of polynomials 23.7-1 ValuePol 23.7-2 ProductCoeffs 23.7-3 ReduceCoeffs 23.7-4 ReduceCoeffsMod 23.7-5 PowerModCoeffs 23.7-6 ShiftedCoeffs 24 Matrices 24.1 InfoMatrix (Info Class) 24.1-1 InfoMatrix 24.2 Categories of Matrices 24.2-1 IsMatrix 24.2-2 IsOrdinaryMatrix 24.2-3 IsLieMatrix 24.3 Operators for Matrices 24.4 Properties and Attributes of Matrices 24.4-1 DimensionsMat 24.4-2 DefaultFieldOfMatrix 24.4-3 TraceMat 24.4-4 DeterminantMat 24.4-5 DeterminantMatDestructive 24.4-6 DeterminantMatDivFree 24.4-7 IsMonomialMatrix 24.4-8 IsDiagonalMat 24.4-9 IsUpperTriangularMat 24.4-10 IsLowerTriangularMat 24.5 Matrix Constructions 24.5-1 IdentityMat 24.5-2 NullMat 24.5-3 EmptyMatrix 24.5-4 DiagonalMat 24.5-5 PermutationMat 24.5-6 TransposedMatImmutable 24.5-7 TransposedMatDestructive 24.5-8 KroneckerProduct 24.5-9 ReflectionMat 24.5-10 PrintArray 24.6 Random Matrices 24.6-1 RandomMat 24.6-2 RandomInvertibleMat 24.6-3 RandomUnimodularMat 24.7 Matrices Representing Linear Equations and the Gaussian Algorithm 24.7-1 RankMat 24.7-2 TriangulizedMat 24.7-3 TriangulizeMat 24.7-4 NullspaceMat 24.7-5 NullspaceMatDestructive 24.7-6 SolutionMat 24.7-7 SolutionMatDestructive 24.7-8 BaseFixedSpace 24.8 Eigenvectors and eigenvalues 24.8-1 GeneralisedEigenvalues 24.8-2 GeneralisedEigenspaces 24.8-3 Eigenvalues 24.8-4 Eigenspaces 24.8-5 Eigenvectors 24.9 Elementary Divisors 24.9-1 ElementaryDivisorsMat 24.9-2 ElementaryDivisorsTransformationsMat 24.9-3 DiagonalizeMat 24.10 Echelonized Matrices 24.10-1 SemiEchelonMat 24.10-2 SemiEchelonMatDestructive 24.10-3 SemiEchelonMatTransformation 24.10-4 SemiEchelonMats 24.10-5 SemiEchelonMatsDestructive 24.11 Matrices as Basis of a Row Space 24.11-1 BaseMat 24.11-2 BaseMatDestructive 24.11-3 BaseOrthogonalSpaceMat 24.11-4 SumIntersectionMat 24.11-5 BaseSteinitzVectors 24.12 Triangular Matrices 24.12-1 DiagonalOfMat 24.12-2 UpperSubdiagonal 24.12-3 DepthOfUpperTriangularMatrix 24.13 Matrices as Linear Mappings 24.13-1 CharacteristicPolynomial 24.13-2 JordanDecomposition 24.13-3 BlownUpMat 24.13-4 BlownUpVector 24.13-5 CompanionMat 24.14 Matrices over Finite Fields 24.14-1 ImmutableMatrix 24.14-2 ConvertToMatrixRep 24.14-3 ProjectiveOrder 24.14-4 SimultaneousEigenvalues 24.15 Inverse and Nullspace of an Integer Matrix Modulo an Ideal 24.15-1 InverseMatMod 24.15-2 NullspaceModQ 24.16 Special Multiplication Algorithms for Matrices over GF(2) 24.16-1 PROD_GF2MAT_GF2MAT_SIMPLE 24.16-2 PROD_GF2MAT_GF2MAT_ADVANCED 24.17 Block Matrices 24.17-1 AsBlockMatrix 24.17-2 BlockMatrix 24.17-3 MatrixByBlockMatrix 25 Integral matrices and lattices 25.1 Linear equations over the integers and Integral Matrices 25.1-1 NullspaceIntMat 25.1-2 SolutionIntMat 25.1-3 SolutionNullspaceIntMat 25.1-4 BaseIntMat 25.1-5 BaseIntersectionIntMats 25.1-6 ComplementIntMat 25.2 Normal Forms over the Integers 25.2-1 TriangulizedIntegerMat 25.2-2 TriangulizedIntegerMatTransform 25.2-3 TriangulizeIntegerMat 25.2-4 HermiteNormalFormIntegerMat 25.2-5 HermiteNormalFormIntegerMatTransform 25.2-6 SmithNormalFormIntegerMat 25.2-7 SmithNormalFormIntegerMatTransforms 25.2-8 DiagonalizeIntMat 25.2-9 NormalFormIntMat 25.2-10 AbelianInvariantsOfList 25.3 Determinant of an integer matrix 25.3-1 DeterminantIntMat 25.4 Decompositions 25.4-1 Decomposition 25.4-2 LinearIndependentColumns 25.4-3 PadicCoefficients 25.4-4 IntegralizedMat 25.4-5 DecompositionInt 25.5 Lattice Reduction 25.5-1 LLLReducedBasis 25.5-2 LLLReducedGramMat 25.6 Orthogonal Embeddings 25.6-1 OrthogonalEmbeddings 25.6-2 ShortestVectors 26 Vector and matrix objects 26.1 Fundamental ideas and rules 26.2 Categories of vectors and matrices 26.3 Constructing vector and matrix objects 26.4 Operations for row vector objects 26.5 Operations for row list matrix objects 26.6 Operations for flat matrix objects 27 Strings and Characters 27.1 IsChar and IsString 27.1-1 IsChar 27.1-2 IsString 27.1-3 Strings As Lists 27.1-4 Printing Strings 27.2 Special Characters 27.3 Triple Quoted Strings 27.4 Internally Represented Strings 27.4-1 IsStringRep 27.4-2 ConvertToStringRep 27.4-3 CopyToStringRep 27.4-4 IsEmptyString 27.4-5 EmptyString 27.4-6 CharsFamily 27.5 Recognizing Characters 27.5-1 IsDigitChar 27.5-2 IsLowerAlphaChar 27.5-3 IsUpperAlphaChar 27.5-4 IsAlphaChar 27.6 Comparisons of Strings 27.6-1 \= 27.6-2 \< 27.7 Operations to Produce or Manipulate Strings 27.7-1 DisplayString 27.7-2 DEFAULTDISPLAYSTRING 27.7-3 ViewString 27.7-4 DEFAULTVIEWSTRING 27.7-5 PrintString 27.7-6 String 27.7-7 StripLineBreakCharacters 27.7-8 HexStringInt 27.7-9 StringPP 27.7-10 WordAlp 27.7-11 LowercaseString 27.7-12 SplitString 27.7-13 ReplacedString 27.7-14 NormalizeWhitespace 27.7-15 NormalizedWhitespace 27.7-16 RemoveCharacters 27.7-17 JoinStringsWithSeparator 27.7-18 Chomp 27.7-19 StartsWith 27.7-20 NumbersString 27.7-21 StringNumbers 27.8 Character Conversion 27.8-1 IntChar 27.8-2 CharInt 27.8-3 SIntChar 27.8-4 CharSInt 27.9 Operations to Evaluate Strings 27.9-1 Int 27.9-2 Ordinal 27.9-3 EvalString 27.9-4 CrcString 27.10 Calendar Arithmetic 27.10-1 DaysInYear 27.10-2 DaysInMonth 27.10-3 DMYDay 27.10-4 DayDMY 27.10-5 WeekDay 27.10-6 StringDate 27.10-7 HMSMSec 27.10-8 SecHMSM 27.10-9 StringTime 27.10-10 SecondsDMYhms 27.10-11 DMYhmsSeconds 27.11 Obtaining LaTeX Representations of Objects 28 Dictionaries and General Hash Tables 28.1 Using Dictionaries 28.2 Dictionaries 28.2-1 NewDictionary 28.3 Dictionaries via Binary Lists 28.3-1 DictionaryByPosition 28.3-2 IsDictionary 28.3-3 IsLookupDictionary 28.3-4 AddDictionary 28.3-5 KnowsDictionary 28.3-6 LookupDictionary 28.4 General Hash Tables 28.5 Hash keys 28.5-1 DenseIntKey 28.5-2 SparseIntKey 28.6 Dense hash tables 28.6-1 DenseHashTable 28.7 Sparse hash tables 28.7-1 SparseHashTable 28.7-2 DoubleHashArraySize 29 Records 29.1 IsRecord and RecNames 29.1-1 IsRecord 29.1-2 RecNames 29.2 Accessing Record Elements 29.3 Record Assignment 29.4 Identical Records 29.5 Comparisons of Records 29.6 IsBound and Unbind for Records 29.6-1 IsBound 29.6-2 Unbind 29.7 Record Access Operations 29.7-1 NameRNam 29.7-2 RNamObj 29.7-3 \. 30 Collections 30.1 IsCollection (Filter) 30.1-1 IsCollection 30.2 Collection Families 30.2-1 CollectionsFamily 30.2-2 IsCollectionFamily 30.2-3 ElementsFamily 30.2-4 CategoryCollections 30.3 Lists and Collections 30.3-1 IsListOrCollection 30.3-2 Enumerator 30.3-3 EnumeratorSorted 30.3-4 EnumeratorByFunctions 30.3-5 List 30.3-6 SortedList 30.3-7 SSortedList 30.3-8 AsList 30.3-9 AsSortedList 30.3-10 AsSSortedList 30.3-11 Elements 30.4 Attributes and Properties for Collections 30.4-1 IsEmpty 30.4-2 IsFinite 30.4-3 IsTrivial 30.4-4 IsNonTrivial 30.4-5 IsWholeFamily 30.4-6 Size 30.4-7 Representative 30.4-8 RepresentativeSmallest 30.5 Operations for Collections 30.5-1 IsSubset 30.5-2 Intersection 30.5-3 Union 30.5-4 Difference 30.6 Membership Test for Collections 30.6-1 \in 30.7 Random Elements 30.7-1 Random 30.7-2 PseudoRandom 30.7-3 RandomList 30.8 Iterators 30.8-1 Iterator 30.8-2 IteratorSorted 30.8-3 IsIterator 30.8-4 IsDoneIterator 30.8-5 NextIterator 30.8-6 IteratorList 30.8-7 TrivialIterator 30.8-8 IteratorByFunctions 31 Domains and their Elements 31.1 Operational Structure of Domains 31.2 Equality and Comparison of Domains 31.3 Constructing Domains 31.4 Changing the Structure 31.5 Changing the Representation 31.6 Domain Categories 31.7 Parents 31.7-1 Parent 31.8 Constructing Subdomains 31.9 Operations for Domains 31.9-1 IsGeneralizedDomain 31.9-2 GeneratorsOfDomain 31.9-3 Domain 31.10 Attributes and Properties of Elements 31.10-1 Characteristic 31.10-2 OneImmutable 31.10-3 ZeroImmutable 31.10-4 MultiplicativeZeroOp 31.10-5 IsOne 31.10-6 IsZero 31.10-7 IsIdempotent 31.10-8 InverseImmutable 31.10-9 AdditiveInverseImmutable 31.10-10 Order 31.11 Comparison Operations for Elements 31.11-1 \= and \< 31.11-2 CanEasilyCompareElements 31.12 Arithmetic Operations for Elements 31.12-1 \+, \*, \/, \^, \mod 31.12-2 LeftQuotient 31.12-3 Comm 31.12-4 LieBracket 31.12-5 Sqrt 31.13 Relations Between Domains 31.13-1 UseSubsetRelation 31.13-2 UseFactorRelation 31.13-3 UseIsomorphismRelation 31.13-4 InstallSubsetMaintenance 31.13-5 InstallFactorMaintenance 31.13-6 InstallIsomorphismMaintenance 31.14 Useful Categories of Elements 31.14-1 IsExtAElement 31.14-2 IsNearAdditiveElement 31.14-3 IsAdditiveElement 31.14-4 IsNearAdditiveElementWithZero 31.14-5 IsAdditiveElementWithZero 31.14-6 IsNearAdditiveElementWithInverse 31.14-7 IsAdditiveElementWithInverse 31.14-8 IsExtLElement 31.14-9 IsExtRElement 31.14-10 IsMultiplicativeElement 31.14-11 IsMultiplicativeElementWithOne 31.14-12 IsMultiplicativeElementWithZero 31.14-13 IsMultiplicativeElementWithInverse 31.14-14 IsVector 31.14-15 IsNearRingElement 31.14-16 IsRingElement 31.14-17 IsNearRingElementWithOne 31.14-18 IsRingElementWithOne 31.14-19 IsNearRingElementWithInverse 31.14-20 IsRingElementWithInverse 31.15 Useful Categories for all Elements of a Family 31.15-1 IsAssociativeElement 31.15-2 IsAdditivelyCommutativeElement 31.15-3 IsCommutativeElement 31.15-4 IsFiniteOrderElement 31.15-5 IsJacobianElement 31.15-6 IsZeroSquaredElement 32 Mappings 32.1 IsDirectProductElement (Filter) 32.1-1 IsDirectProductElement 32.2 Creating Mappings 32.2-1 GeneralMappingByElements 32.2-2 MappingByFunction 32.2-3 InverseGeneralMapping 32.2-4 CompositionMapping 32.2-5 CompositionMapping2 32.2-6 IsCompositionMappingRep 32.2-7 ConstituentsCompositionMapping 32.2-8 ZeroMapping 32.2-9 IdentityMapping 32.2-10 Embedding 32.2-11 Projection 32.2-12 RestrictedMapping 32.3 Properties and Attributes of (General) Mappings 32.3-1 IsTotal 32.3-2 IsSingleValued 32.3-3 IsMapping 32.3-4 IsInjective 32.3-5 IsSurjective 32.3-6 IsBijective 32.3-7 Range 32.3-8 Source 32.3-9 UnderlyingRelation 32.3-10 UnderlyingGeneralMapping 32.4 Images under Mappings 32.4-1 ImagesSource 32.4-2 ImagesRepresentative 32.4-3 ImagesElm 32.4-4 ImagesSet 32.4-5 ImageElm 32.4-6 Image 32.4-7 Images 32.5 Preimages under Mappings 32.5-1 PreImagesRange 32.5-2 PreImagesElm 32.5-3 PreImageElm 32.5-4 PreImagesRepresentative 32.5-5 PreImagesSet 32.5-6 PreImage 32.5-7 PreImages 32.6 Arithmetic Operations for General Mappings 32.7 Mappings which are Compatible with Algebraic Structures 32.8 Magma Homomorphisms 32.8-1 IsMagmaHomomorphism 32.8-2 MagmaHomomorphismByFunctionNC 32.8-3 NaturalHomomorphismByGenerators 32.9 Mappings that Respect Multiplication 32.9-1 RespectsMultiplication 32.9-2 RespectsOne 32.9-3 RespectsInverses 32.9-4 IsGroupGeneralMapping 32.9-5 KernelOfMultiplicativeGeneralMapping 32.9-6 CoKernelOfMultiplicativeGeneralMapping 32.10 Mappings that Respect Addition 32.10-1 RespectsAddition 32.10-2 RespectsAdditiveInverses 32.10-3 RespectsZero 32.10-4 IsAdditiveGroupGeneralMapping 32.10-5 KernelOfAdditiveGeneralMapping 32.10-6 CoKernelOfAdditiveGeneralMapping 32.11 Linear Mappings 32.11-1 RespectsScalarMultiplication 32.11-2 IsLeftModuleGeneralMapping 32.11-3 IsLinearMapping 32.12 Ring Homomorphisms 32.12-1 IsRingGeneralMapping 32.12-2 IsRingWithOneGeneralMapping 32.12-3 IsAlgebraGeneralMapping 32.12-4 IsAlgebraWithOneGeneralMapping 32.12-5 IsFieldHomomorphism 32.13 General Mappings 32.13-1 IsGeneralMapping 32.13-2 IsConstantTimeAccessGeneralMapping 32.13-3 IsEndoGeneralMapping 32.14 Technical Matters Concerning General Mappings 32.14-1 IsSPGeneralMapping 32.14-2 IsGeneralMappingFamily 32.14-3 FamilyRange 32.14-4 FamilySource 32.14-5 FamiliesOfGeneralMappingsAndRanges 32.14-6 GeneralMappingsFamily 32.14-7 TypeOfDefaultGeneralMapping 33 Relations 33.1 General Binary Relations 33.1-1 IsBinaryRelation 33.1-2 BinaryRelationByElements 33.1-3 IdentityBinaryRelation 33.1-4 EmptyBinaryRelation 33.2 Properties and Attributes of Binary Relations 33.2-1 IsReflexiveBinaryRelation 33.2-2 IsSymmetricBinaryRelation 33.2-3 IsTransitiveBinaryRelation 33.2-4 IsAntisymmetricBinaryRelation 33.2-5 IsPreOrderBinaryRelation 33.2-6 IsPartialOrderBinaryRelation 33.2-7 IsHasseDiagram 33.2-8 IsEquivalenceRelation 33.2-9 Successors 33.2-10 DegreeOfBinaryRelation 33.2-11 PartialOrderOfHasseDiagram 33.3 Binary Relations on Points 33.3-1 BinaryRelationOnPoints 33.3-2 RandomBinaryRelationOnPoints 33.3-3 AsBinaryRelationOnPoints 33.4 Closure Operations and Other Constructors 33.4-1 ReflexiveClosureBinaryRelation 33.4-2 SymmetricClosureBinaryRelation 33.4-3 TransitiveClosureBinaryRelation 33.4-4 HasseDiagramBinaryRelation 33.4-5 StronglyConnectedComponents 33.4-6 PartialOrderByOrderingFunction 33.5 Equivalence Relations 33.5-1 EquivalenceRelationByPartition 33.5-2 EquivalenceRelationByRelation 33.5-3 EquivalenceRelationByPairs 33.5-4 EquivalenceRelationByProperty 33.6 Attributes of and Operations on Equivalence Relations 33.6-1 EquivalenceRelationPartition 33.6-2 GeneratorsOfEquivalenceRelationPartition 33.6-3 JoinEquivalenceRelations 33.7 Equivalence Classes 33.7-1 IsEquivalenceClass 33.7-2 EquivalenceClassRelation 33.7-3 EquivalenceClasses 33.7-4 EquivalenceClassOfElement 34 Orderings 34.1 IsOrdering (Filter) 34.1-1 IsOrdering 34.1-2 OrderingsFamily 34.2 Building new orderings 34.2-1 OrderingByLessThanFunctionNC 34.2-2 OrderingByLessThanOrEqualFunctionNC 34.3 Properties and basic functionality 34.3-1 IsWellFoundedOrdering 34.3-2 IsTotalOrdering 34.3-3 IsIncomparableUnder 34.3-4 FamilyForOrdering 34.3-5 LessThanFunction 34.3-6 LessThanOrEqualFunction 34.3-7 IsLessThanUnder 34.3-8 IsLessThanOrEqualUnder 34.4 Orderings on families of associative words 34.4-1 IsOrderingOnFamilyOfAssocWords 34.4-2 IsTranslationInvariantOrdering 34.4-3 IsReductionOrdering 34.4-4 OrderingOnGenerators 34.4-5 LexicographicOrdering 34.4-6 ShortLexOrdering 34.4-7 IsShortLexOrdering 34.4-8 WeightLexOrdering 34.4-9 IsWeightLexOrdering 34.4-10 WeightOfGenerators 34.4-11 BasicWreathProductOrdering 34.4-12 IsBasicWreathProductOrdering 34.4-13 WreathProductOrdering 34.4-14 IsWreathProductOrdering 34.4-15 LevelsOfGenerators 35 Magmas 35.1 Magma Categories 35.1-1 IsMagma 35.1-2 IsMagmaWithOne 35.1-3 IsMagmaWithInversesIfNonzero 35.1-4 IsMagmaWithInverses 35.2 Magma Generation 35.2-1 Magma 35.2-2 MagmaWithOne 35.2-3 MagmaWithInverses 35.2-4 MagmaByGenerators 35.2-5 MagmaWithOneByGenerators 35.2-6 MagmaWithInversesByGenerators 35.2-7 Submagma 35.2-8 SubmagmaWithOne 35.2-9 SubmagmaWithInverses 35.2-10 AsMagma 35.2-11 AsSubmagma 35.2-12 IsMagmaWithZeroAdjoined 35.2-13 InjectionZeroMagma 35.2-14 UnderlyingInjectionZeroMagma 35.3 Magmas Defined by Multiplication Tables 35.3-1 MagmaByMultiplicationTable 35.3-2 MagmaWithOneByMultiplicationTable 35.3-3 MagmaWithInversesByMultiplicationTable 35.3-4 MagmaElement 35.3-5 MultiplicationTable 35.4 Attributes and Properties for Magmas 35.4-1 GeneratorsOfMagma 35.4-2 GeneratorsOfMagmaWithOne 35.4-3 GeneratorsOfMagmaWithInverses 35.4-4 Centralizer 35.4-5 Centre 35.4-6 Idempotents 35.4-7 IsAssociative 35.4-8 IsCentral 35.4-9 IsCommutative 35.4-10 MultiplicativeNeutralElement 35.4-11 MultiplicativeZero 35.4-12 SquareRoots 35.4-13 TrivialSubmagmaWithOne 36 Words 36.1 Categories of Words and Nonassociative Words 36.1-1 IsWord 36.1-2 IsWordCollection 36.1-3 IsNonassocWord 36.1-4 IsNonassocWordCollection 36.2 Comparison of Words 36.2-1 \= 36.2-2 \< 36.3 Operations for Words 36.3-1 MappedWord 36.4 Free Magmas 36.4-1 FreeMagma 36.4-2 FreeMagmaWithOne 36.5 External Representation for Nonassociative Words 37 Associative Words 37.1 Categories of Associative Words 37.1-1 IsAssocWord 37.2 Free Groups, Monoids and Semigroups 37.2-1 FreeGroup 37.2-2 IsFreeGroup 37.2-3 AssignGeneratorVariables 37.3 Comparison of Associative Words 37.3-1 \= 37.3-2 \< 37.3-3 IsShortLexLessThanOrEqual 37.3-4 IsBasicWreathLessThanOrEqual 37.4 Operations for Associative Words 37.4-1 Length 37.4-2 ExponentSumWord 37.4-3 Subword 37.4-4 PositionWord 37.4-5 SubstitutedWord 37.4-6 EliminatedWord 37.5 Operations for Associative Words by their Syllables 37.5-1 NumberSyllables 37.5-2 ExponentSyllable 37.5-3 GeneratorSyllable 37.5-4 SubSyllables 37.6 Representations for Associative Words 37.6-1 IsLetterAssocWordRep 37.6-2 IsLetterWordsFamily 37.6-3 IsBLetterAssocWordRep 37.6-4 IsBLetterWordsFamily 37.6-5 IsSyllableAssocWordRep 37.6-6 IsSyllableWordsFamily 37.6-7 Is16BitsFamily 37.6-8 LetterRepAssocWord 37.6-9 AssocWordByLetterRep 37.7 The External Representation for Associative Words 37.8 Straight Line Programs 37.8-1 IsStraightLineProgram 37.8-2 StraightLineProgram 37.8-3 LinesOfStraightLineProgram 37.8-4 NrInputsOfStraightLineProgram 37.8-5 ResultOfStraightLineProgram 37.8-6 StringOfResultOfStraightLineProgram 37.8-7 CompositionOfStraightLinePrograms 37.8-8 IntegratedStraightLineProgram 37.8-9 RestrictOutputsOfSLP 37.8-10 IntermediateResultOfSLP 37.8-11 IntermediateResultOfSLPWithoutOverwrite 37.8-12 IntermediateResultsOfSLPWithoutOverwrite 37.8-13 ProductOfStraightLinePrograms 37.8-14 SlotUsagePattern 37.9 Straight Line Program Elements 37.9-1 IsStraightLineProgElm 37.9-2 StraightLineProgElm 37.9-3 StraightLineProgGens 37.9-4 EvalStraightLineProgElm 37.9-5 StretchImportantSLPElement 38 Rewriting Systems 38.1 Operations on rewriting systems 38.1-1 IsRewritingSystem 38.1-2 Rules 38.1-3 OrderOfRewritingSystem 38.1-4 ReducedForm 38.1-5 IsConfluent 38.1-6 ConfluentRws 38.1-7 IsReduced 38.1-8 ReduceRules 38.1-9 AddRule 38.1-10 AddRuleReduced 38.1-11 MakeConfluent 38.1-12 GeneratorsOfRws 38.2 Operations on elements of the algebra 38.2-1 ReducedProduct 38.3 Properties of rewriting systems 38.3-1 IsBuiltFromAdditiveMagmaWithInverses 38.4 Rewriting in Groups and Monoids 38.5 Developing rewriting systems 39 Groups 39.1 Group Elements 39.2 Creating Groups 39.2-1 Group 39.2-2 GroupByGenerators 39.2-3 GroupWithGenerators 39.2-4 GeneratorsOfGroup 39.2-5 AsGroup 39.2-6 ConjugateGroup 39.2-7 IsGroup 39.2-8 InfoGroup 39.3 Subgroups 39.3-1 Subgroup 39.3-2 Index (GAP operation) 39.3-3 IndexInWholeGroup 39.3-4 AsSubgroup 39.3-5 IsSubgroup 39.3-6 IsNormal 39.3-7 IsCharacteristicSubgroup 39.3-8 ConjugateSubgroup 39.3-9 ConjugateSubgroups 39.3-10 IsSubnormal 39.3-11 SubgroupByProperty 39.3-12 SubgroupShell 39.4 Closures of (Sub)groups 39.4-1 ClosureGroup 39.4-2 ClosureGroupAddElm 39.4-3 ClosureGroupDefault 39.4-4 ClosureSubgroup 39.5 Expressing Group Elements as Words in Generators 39.5-1 EpimorphismFromFreeGroup 39.5-2 Factorization 39.5-3 GrowthFunctionOfGroup 39.6 Structure Descriptions 39.6-1 StructureDescription 39.7 Cosets 39.7-1 RightCoset 39.7-2 RightCosets 39.7-3 CanonicalRightCosetElement 39.7-4 IsRightCoset 39.7-5 CosetDecomposition 39.8 Transversals 39.8-1 RightTransversal 39.9 Double Cosets 39.9-1 DoubleCoset 39.9-2 RepresentativesContainedRightCosets 39.9-3 DoubleCosets 39.9-4 IsDoubleCoset 39.9-5 DoubleCosetRepsAndSizes 39.9-6 InfoCoset 39.10 Conjugacy Classes 39.10-1 ConjugacyClass 39.10-2 ConjugacyClasses 39.10-3 ConjugacyClassesByRandomSearch 39.10-4 ConjugacyClassesByOrbits 39.10-5 NrConjugacyClasses 39.10-6 RationalClass 39.10-7 RationalClasses 39.10-8 GaloisGroup 39.10-9 IsConjugate 39.10-10 NthRootsInGroup 39.11 Normal Structure 39.11-1 Normalizer 39.11-2 Core 39.11-3 PCore 39.11-4 NormalClosure 39.11-5 NormalIntersection 39.11-6 ComplementClassesRepresentatives 39.11-7 InfoComplement 39.12 Specific and Parametrized Subgroups 39.12-1 TrivialSubgroup 39.12-2 CommutatorSubgroup 39.12-3 DerivedSubgroup 39.12-4 CommutatorLength 39.12-5 FittingSubgroup 39.12-6 FrattiniSubgroup 39.12-7 PrefrattiniSubgroup 39.12-8 PerfectResiduum 39.12-9 RadicalGroup 39.12-10 Socle 39.12-11 SupersolvableResiduum 39.12-12 PRump 39.13 Sylow Subgroups and Hall Subgroups 39.13-1 SylowSubgroup 39.13-2 SylowComplement 39.13-3 HallSubgroup 39.13-4 SylowSystem 39.13-5 ComplementSystem 39.13-6 HallSystem 39.14 Subgroups characterized by prime powers 39.14-1 Omega 39.14-2 Agemo 39.15 Group Properties 39.15-1 IsCyclic 39.15-2 IsElementaryAbelian 39.15-3 IsNilpotentGroup 39.15-4 NilpotencyClassOfGroup 39.15-5 IsPerfectGroup 39.15-6 IsSolvableGroup 39.15-7 IsPolycyclicGroup 39.15-8 IsSupersolvableGroup 39.15-9 IsMonomialGroup 39.15-10 IsSimpleGroup 39.15-11 IsAlmostSimpleGroup 39.15-12 IsomorphismTypeInfoFiniteSimpleGroup 39.15-13 SimpleGroup 39.15-14 SimpleGroupsIterator 39.15-15 SmallSimpleGroup 39.15-16 AllSmallNonabelianSimpleGroups 39.15-17 IsFinitelyGeneratedGroup 39.15-18 IsSubsetLocallyFiniteGroup 39.15-19 IsPGroup 39.15-20 PrimePGroup 39.15-21 PClassPGroup 39.15-22 RankPGroup 39.15-23 IsPSolvable 39.15-24 IsPNilpotent 39.16 Numerical Group Attributes 39.16-1 AbelianInvariants 39.16-2 Exponent 39.16-3 EulerianFunction 39.17 Subgroup Series 39.17-1 ChiefSeries 39.17-2 ChiefSeriesThrough 39.17-3 ChiefSeriesUnderAction 39.17-4 SubnormalSeries 39.17-5 CompositionSeries 39.17-6 DisplayCompositionSeries 39.17-7 DerivedSeriesOfGroup 39.17-8 DerivedLength 39.17-9 ElementaryAbelianSeries 39.17-10 InvariantElementaryAbelianSeries 39.17-11 LowerCentralSeriesOfGroup 39.17-12 UpperCentralSeriesOfGroup 39.17-13 PCentralSeries 39.17-14 JenningsSeries 39.17-15 DimensionsLoewyFactors 39.17-16 AscendingChain 39.17-17 IntermediateGroup 39.17-18 IntermediateSubgroups 39.18 Factor Groups 39.18-1 NaturalHomomorphismByNormalSubgroup 39.18-2 FactorGroup 39.18-3 CommutatorFactorGroup 39.18-4 MaximalAbelianQuotient 39.18-5 HasAbelianFactorGroup 39.18-6 HasElementaryAbelianFactorGroup 39.18-7 CentralizerModulo 39.19 Sets of Subgroups 39.19-1 ConjugacyClassSubgroups 39.19-2 IsConjugacyClassSubgroupsRep 39.19-3 ConjugacyClassesSubgroups 39.19-4 ConjugacyClassesMaximalSubgroups 39.19-5 AllSubgroups 39.19-6 MaximalSubgroupClassReps 39.19-7 MaximalSubgroups 39.19-8 NormalSubgroups 39.19-9 MaximalNormalSubgroups 39.19-10 MinimalNormalSubgroups 39.20 Subgroup Lattice 39.20-1 LatticeSubgroups 39.20-2 ClassElementLattice 39.20-3 DotFileLatticeSubgroups 39.20-4 MaximalSubgroupsLattice 39.20-5 MinimalSupergroupsLattice 39.20-6 RepresentativesPerfectSubgroups 39.20-7 ConjugacyClassesPerfectSubgroups 39.20-8 Zuppos 39.20-9 InfoLattice 39.21 Specific Methods for Subgroup Lattice Computations 39.21-1 LatticeByCyclicExtension 39.21-2 InvariantSubgroupsElementaryAbelianGroup 39.21-3 SubgroupsSolvableGroup 39.21-4 SizeConsiderFunction 39.21-5 ExactSizeConsiderFunction 39.21-6 InfoPcSubgroup 39.22 Special Generating Sets 39.22-1 GeneratorsSmallest 39.22-2 LargestElementGroup 39.22-3 MinimalGeneratingSet 39.22-4 SmallGeneratingSet 39.22-5 IndependentGeneratorsOfAbelianGroup 39.22-6 IndependentGeneratorExponents 39.23 1-Cohomology 39.23-1 OneCocycles 39.23-2 OneCoboundaries 39.23-3 OCOneCocycles 39.23-4 ComplementClassesRepresentativesEA 39.23-5 InfoCoh 39.24 Schur Covers and Multipliers 39.24-1 EpimorphismSchurCover 39.24-2 SchurCover 39.24-3 AbelianInvariantsMultiplier 39.24-4 Epicentre 39.24-5 NonabelianExteriorSquare 39.24-6 EpimorphismNonabelianExteriorSquare 39.24-7 IsCentralFactor 39.24-8 Covering groups of symmetric groups 39.24-9 BasicSpinRepresentationOfSymmetricGroup 39.24-10 SchurCoverOfSymmetricGroup 39.24-11 DoubleCoverOfAlternatingGroup 39.25 Tests for the Availability of Methods 39.25-1 CanEasilyTestMembership 39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup 39.25-3 CanComputeSize 39.25-4 CanComputeSizeAnySubgroup 39.25-5 CanComputeIndex 39.25-6 CanComputeIsSubset 39.25-7 KnowsHowToDecompose 40 Group Homomorphisms 40.1 Creating Group Homomorphisms 40.1-1 GroupHomomorphismByImages 40.1-2 GroupHomomorphismByImagesNC 40.1-3 GroupGeneralMappingByImages 40.1-4 GroupHomomorphismByFunction 40.1-5 AsGroupGeneralMappingByImages 40.2 Operations for Group Homomorphisms 40.3 Efficiency of Homomorphisms 40.3-1 Mappings given on generators 40.3-2 Action homomorphisms 40.3-3 Mappings given by functions 40.3-4 Other operations 40.3-5 ImagesSmallestGenerators 40.4 Homomorphism for very large groups 40.5 Nice Monomorphisms 40.5-1 IsHandledByNiceMonomorphism 40.5-2 NiceMonomorphism 40.5-3 NiceObject 40.5-4 IsCanonicalNiceMonomorphism 40.6 Group Automorphisms 40.6-1 ConjugatorIsomorphism 40.6-2 ConjugatorAutomorphism 40.6-3 InnerAutomorphism 40.6-4 IsConjugatorIsomorphism 40.6-5 ConjugatorOfConjugatorIsomorphism 40.7 Groups of Automorphisms 40.7-1 AutomorphismGroup 40.7-2 IsGroupOfAutomorphisms 40.7-3 AutomorphismDomain 40.7-4 IsAutomorphismGroup 40.7-5 InnerAutomorphismsAutomorphismGroup 40.7-6 InducedAutomorphism 40.8 Calculating with Group Automorphisms 40.8-1 AssignNiceMonomorphismAutomorphismGroup 40.8-2 NiceMonomorphismAutomGroup 40.9 Searching for Homomorphisms 40.9-1 IsomorphismGroups 40.9-2 AllHomomorphismClasses 40.9-3 AllHomomorphisms 40.9-4 GQuotients 40.9-5 IsomorphicSubgroups 40.9-6 MorClassLoop 40.10 Representations for Group Homomorphisms 40.10-1 IsGroupGeneralMappingByImages 40.10-2 MappingGeneratorsImages 40.10-3 IsGroupGeneralMappingByAsGroupGeneralMappingByImages 40.10-4 IsPreimagesByAsGroupGeneralMappingByImages 40.10-5 IsPermGroupGeneralMapping 40.10-6 IsToPermGroupGeneralMappingByImages 40.10-7 IsGroupGeneralMappingByPcgs 40.10-8 IsPcGroupGeneralMappingByImages 40.10-9 IsToPcGroupGeneralMappingByImages 40.10-10 IsFromFpGroupGeneralMappingByImages 40.10-11 IsFromFpGroupStdGensGeneralMappingByImages 41 Group Actions 41.1 About Group Actions 41.2 Basic Actions 41.2-1 OnPoints 41.2-2 OnRight 41.2-3 OnLeftInverse 41.2-4 OnSets 41.2-5 OnTuples 41.2-6 OnPairs 41.2-7 OnSetsSets 41.2-8 OnSetsDisjointSets 41.2-9 OnSetsTuples 41.2-10 OnTuplesSets 41.2-11 OnTuplesTuples 41.2-12 OnLines 41.2-13 OnIndeterminates 41.2-14 Permuted 41.2-15 OnSubspacesByCanonicalBasis 41.3 Action on canonical representatives 41.4 Orbits 41.4-1 Orbit 41.4-2 Orbits 41.4-3 OrbitsDomain 41.4-4 OrbitLength 41.4-5 OrbitLengths 41.4-6 OrbitLengthsDomain 41.5 Stabilizers 41.5-1 OrbitStabilizer 41.5-2 Stabilizer 41.5-3 OrbitStabilizerAlgorithm 41.6 Elements with Prescribed Images 41.6-1 RepresentativeAction 41.7 The Permutation Image of an Action 41.7-1 ActionHomomorphism 41.7-2 Action 41.7-3 SparseActionHomomorphism 41.8 Action of a group on itself 41.8-1 FactorCosetAction 41.8-2 RegularActionHomomorphism 41.8-3 AbelianSubfactorAction 41.9 Permutations Induced by Elements and Cycles 41.9-1 Permutation 41.9-2 PermutationCycle 41.9-3 Cycle 41.9-4 CycleLength 41.9-5 Cycles 41.9-6 CycleLengths 41.9-7 CycleIndex 41.10 Tests for Actions 41.10-1 IsTransitive 41.10-2 Transitivity 41.10-3 RankAction 41.10-4 IsSemiRegular 41.10-5 IsRegular 41.10-6 Earns 41.10-7 IsPrimitive 41.11 Block Systems 41.11-1 Blocks 41.11-2 MaximalBlocks 41.11-3 RepresentativesMinimalBlocks 41.11-4 AllBlocks 41.12 External Sets 41.12-1 IsExternalSet 41.12-2 ExternalSet 41.12-3 ActingDomain 41.12-4 FunctionAction 41.12-5 HomeEnumerator 41.12-6 IsExternalSubset 41.12-7 ExternalSubset 41.12-8 IsExternalOrbit 41.12-9 ExternalOrbit 41.12-10 StabilizerOfExternalSet 41.12-11 ExternalOrbits 41.12-12 ExternalOrbitsStabilizers 41.12-13 CanonicalRepresentativeOfExternalSet 41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet 41.12-15 ActorOfExternalSet 41.12-16 UnderlyingExternalSet 41.12-17 SurjectiveActionHomomorphismAttr 42 Permutations 42.1 IsPerm (Filter) 42.1-1 IsPerm 42.1-2 IsPermCollection 42.1-3 PermutationsFamily 42.2 Comparison of Permutations 42.2-1 \= 42.2-2 DistancePerms 42.2-3 SmallestGeneratorPerm 42.3 Moved Points of Permutations 42.3-1 SmallestMovedPoint 42.3-2 LargestMovedPoint 42.3-3 MovedPoints 42.3-4 NrMovedPoints 42.4 Sign and Cycle Structure 42.4-1 SignPerm 42.4-2 CycleStructurePerm 42.5 Creating Permutations 42.5-1 ListPerm 42.5-2 PermList 42.5-3 MappingPermListList 42.5-4 RestrictedPerm 42.5-5 AsPermutation 43 Permutation Groups 43.1 IsPermGroup (Filter) 43.1-1 IsPermGroup 43.2 The Natural Action 43.2-1 OrbitPerms 43.2-2 OrbitsPerms 43.3 Computing a Permutation Representation 43.3-1 IsomorphismPermGroup 43.3-2 SmallerDegreePermutationRepresentation 43.4 Symmetric and Alternating Groups 43.4-1 IsNaturalSymmetricGroup 43.4-2 IsSymmetricGroup 43.4-3 IsAlternatingGroup 43.4-4 SymmetricParentGroup 43.5 Primitive Groups 43.5-1 ONanScottType 43.5-2 SocleTypePrimitiveGroup 43.6 Stabilizer Chains 43.7 Randomized Methods for Permutation Groups 43.8 Construction of Stabilizer Chains 43.8-1 StabChain 43.8-2 StabChainOptions 43.8-3 DefaultStabChainOptions 43.8-4 StabChainBaseStrongGenerators 43.8-5 MinimalStabChain 43.9 Stabilizer Chain Records 43.10 Operations for Stabilizer Chains 43.10-1 BaseStabChain 43.10-2 BaseOfGroup 43.10-3 SizeStabChain 43.10-4 StrongGeneratorsStabChain 43.10-5 GroupStabChain 43.10-6 OrbitStabChain 43.10-7 IndicesStabChain 43.10-8 ListStabChain 43.10-9 ElementsStabChain 43.10-10 IteratorStabChain 43.10-11 InverseRepresentative 43.10-12 SiftedPermutation 43.10-13 MinimalElementCosetStabChain 43.10-14 LargestElementStabChain 43.10-15 ApproximateSuborbitsStabilizerPermGroup 43.11 Low Level Routines to Modify and Create Stabilizer Chains 43.11-1 CopyStabChain 43.11-2 CopyOptionsDefaults 43.11-3 ChangeStabChain 43.11-4 ExtendStabChain 43.11-5 ReduceStabChain 43.11-6 RemoveStabChain 43.11-7 EmptyStabChain 43.11-8 InsertTrivialStabilizer 43.11-9 IsFixedStabilizer 43.11-10 AddGeneratorsExtendSchreierTree 43.12 Backtrack 43.12-1 SubgroupProperty 43.12-2 ElementProperty 43.12-3 TwoClosure 43.12-4 InfoBckt 43.13 Working with large degree permutation groups 44 Matrix Groups 44.1 IsMatrixGroup (Filter) 44.1-1 IsMatrixGroup 44.2 Attributes and Properties for Matrix Groups 44.2-1 DimensionOfMatrixGroup 44.2-2 DefaultFieldOfMatrixGroup 44.2-3 FieldOfMatrixGroup 44.2-4 TransposedMatrixGroup 44.2-5 IsFFEMatrixGroup 44.3 Actions of Matrix Groups 44.3-1 ProjectiveActionOnFullSpace 44.3-2 ProjectiveActionHomomorphismMatrixGroup 44.3-3 BlowUpIsomorphism 44.4 GL and SL 44.4-1 IsGeneralLinearGroup 44.4-2 IsNaturalGL 44.4-3 IsSpecialLinearGroup 44.4-4 IsNaturalSL 44.4-5 IsSubgroupSL 44.5 Invariant Forms 44.5-1 InvariantBilinearForm 44.5-2 IsFullSubgroupGLorSLRespectingBilinearForm 44.5-3 InvariantSesquilinearForm 44.5-4 IsFullSubgroupGLorSLRespectingSesquilinearForm 44.5-5 InvariantQuadraticForm 44.5-6 IsFullSubgroupGLorSLRespectingQuadraticForm 44.6 Matrix Groups in Characteristic 0 44.6-1 IsCyclotomicMatrixGroup 44.6-2 IsRationalMatrixGroup 44.6-3 IsIntegerMatrixGroup 44.6-4 IsNaturalGLnZ 44.6-5 IsNaturalSLnZ 44.6-6 InvariantLattice 44.6-7 NormalizerInGLnZ 44.6-8 CentralizerInGLnZ 44.6-9 ZClassRepsQClass 44.6-10 IsBravaisGroup 44.6-11 BravaisGroup 44.6-12 BravaisSubgroups 44.6-13 BravaisSupergroups 44.6-14 NormalizerInGLnZBravaisGroup 44.7 Acting OnRight and OnLeft 44.7-1 CrystGroupDefaultAction 44.7-2 SetCrystGroupDefaultAction 45 Polycyclic Groups 45.1 Polycyclic Generating Systems 45.2 Computing a Pcgs 45.2-1 Pcgs 45.2-2 IsPcgs 45.2-3 CanEasilyComputePcgs 45.3 Defining a Pcgs Yourself 45.3-1 PcgsByPcSequence 45.4 Elementary Operations for a Pcgs 45.4-1 RelativeOrders 45.4-2 IsFiniteOrdersPcgs 45.4-3 IsPrimeOrdersPcgs 45.4-4 PcSeries 45.4-5 GroupOfPcgs 45.4-6 OneOfPcgs 45.5 Elementary Operations for a Pcgs and an Element 45.5-1 RelativeOrderOfPcElement 45.5-2 ExponentOfPcElement 45.5-3 ExponentsOfPcElement 45.5-4 DepthOfPcElement 45.5-5 LeadingExponentOfPcElement 45.5-6 PcElementByExponents 45.5-7 LinearCombinationPcgs 45.5-8 SiftedPcElement 45.5-9 CanonicalPcElement 45.5-10 ReducedPcElement 45.5-11 CleanedTailPcElement 45.5-12 HeadPcElementByNumber 45.6 Exponents of Special Products 45.6-1 ExponentsConjugateLayer 45.6-2 ExponentsOfRelativePower 45.6-3 ExponentsOfConjugate 45.6-4 ExponentsOfCommutator 45.7 Subgroups of Polycyclic Groups - Induced Pcgs 45.7-1 IsInducedPcgs 45.7-2 InducedPcgsByPcSequence 45.7-3 ParentPcgs 45.7-4 InducedPcgs 45.7-5 InducedPcgsByGenerators 45.7-6 InducedPcgsByPcSequenceAndGenerators 45.7-7 LeadCoeffsIGS 45.7-8 ExtendedPcgs 45.7-9 SubgroupByPcgs 45.8 Subgroups of Polycyclic Groups – Canonical Pcgs 45.8-1 IsCanonicalPcgs 45.8-2 CanonicalPcgs 45.9 Factor Groups of Polycyclic Groups – Modulo Pcgs 45.9-1 ModuloPcgs 45.9-2 IsModuloPcgs 45.9-3 NumeratorOfModuloPcgs 45.9-4 DenominatorOfModuloPcgs 45.9-5 \mod 45.9-6 CorrespondingGeneratorsByModuloPcgs 45.9-7 CanonicalPcgsByGeneratorsWithImages 45.10 Factor Groups of Polycyclic Groups in their Own Representation 45.10-1 ProjectedPcElement 45.10-2 ProjectedInducedPcgs 45.10-3 LiftedPcElement 45.10-4 LiftedInducedPcgs 45.11 Pcgs and Normal Series 45.11-1 IsPcgsElementaryAbelianSeries 45.11-2 PcgsElementaryAbelianSeries 45.11-3 IndicesEANormalSteps 45.11-4 EANormalSeriesByPcgs 45.11-5 IsPcgsCentralSeries 45.11-6 PcgsCentralSeries 45.11-7 IndicesCentralNormalSteps 45.11-8 CentralNormalSeriesByPcgs 45.11-9 IsPcgsPCentralSeriesPGroup 45.11-10 PcgsPCentralSeriesPGroup 45.11-11 IndicesPCentralNormalStepsPGroup 45.11-12 PCentralNormalSeriesByPcgsPGroup 45.11-13 IsPcgsChiefSeries 45.11-14 PcgsChiefSeries 45.11-15 IndicesChiefNormalSteps 45.11-16 ChiefNormalSeriesByPcgs 45.11-17 IndicesNormalSteps 45.11-18 NormalSeriesByPcgs 45.12 Sum and Intersection of Pcgs 45.12-1 SumFactorizationFunctionPcgs 45.13 Special Pcgs 45.13-1 IsSpecialPcgs 45.13-2 SpecialPcgs 45.13-3 LGWeights 45.13-4 LGLayers 45.13-5 LGFirst 45.13-6 LGLength 45.13-7 IsInducedPcgsWrtSpecialPcgs 45.13-8 InducedPcgsWrtSpecialPcgs 45.14 Action on Subfactors Defined by a Pcgs 45.14-1 VectorSpaceByPcgsOfElementaryAbelianGroup 45.14-2 LinearAction 45.14-3 LinearActionLayer 45.14-4 AffineAction 45.14-5 AffineActionLayer 45.15 Orbit Stabilizer Methods for Polycyclic Groups 45.15-1 StabilizerPcgs 45.15-2 Pcgs_OrbitStabilizer 45.16 Operations which have Special Methods for Groups with Pcgs 45.17 Conjugacy Classes in Solvable Groups 45.17-1 ClassesSolvableGroup 45.17-2 CentralizerSizeLimitConsiderFunction 46 Pc Groups 46.1 The family pcgs 46.1-1 FamilyPcgs 46.1-2 IsFamilyPcgs 46.1-3 InducedPcgsWrtFamilyPcgs 46.1-4 IsParentPcgsFamilyPcgs 46.2 Elements of pc groups 46.2-1 Comparison of elements of pc groups 46.2-2 Arithmetic operations for elements of pc groups 46.3 Pc groups versus fp groups 46.3-1 IsPcGroup 46.3-2 IsomorphismFpGroupByPcgs 46.4 Constructing Pc Groups 46.4-1 PcGroupFpGroup 46.4-2 SingleCollector 46.4-3 SetConjugate 46.4-4 SetCommutator 46.4-5 SetPower 46.4-6 GroupByRws 46.4-7 IsConfluent 46.4-8 IsomorphismRefinedPcGroup 46.4-9 RefinedPcGroup 46.5 Computing Pc Groups 46.5-1 PcGroupWithPcgs 46.5-2 IsomorphismPcGroup 46.5-3 IsomorphismSpecialPcGroup 46.6 Saving a Pc Group 46.6-1 GapInputPcGroup 46.7 Operations for Pc Groups 46.8 2-Cohomology and Extensions 46.8-1 TwoCoboundaries 46.8-2 TwoCocycles 46.8-3 TwoCohomology 46.8-4 Extensions 46.8-5 Extension 46.8-6 SplitExtension 46.8-7 ModuleOfExtension 46.8-8 CompatiblePairs 46.8-9 ExtensionRepresentatives 46.8-10 SplitExtension 46.9 Coding a Pc Presentation 46.9-1 CodePcgs 46.9-2 CodePcGroup 46.9-3 PcGroupCode 46.10 Random Isomorphism Testing 46.10-1 RandomIsomorphismTest 47 Finitely Presented Groups 47.1 IsSubgroupFpGroup and IsFpGroup 47.1-1 IsSubgroupFpGroup 47.1-2 IsFpGroup 47.1-3 InfoFpGroup 47.2 Creating Finitely Presented Groups 47.2-1 \/ 47.2-2 FactorGroupFpGroupByRels 47.2-3 ParseRelators 47.2-4 StringFactorizationWord 47.3 Comparison of Elements of Finitely Presented Groups 47.3-1 \= 47.3-2 \< 47.3-3 FpElmComparisonMethod 47.3-4 SetReducedMultiplication 47.4 Preimages in the Free Group 47.4-1 FreeGroupOfFpGroup 47.4-2 FreeGeneratorsOfFpGroup 47.4-3 RelatorsOfFpGroup 47.4-4 UnderlyingElement 47.4-5 ElementOfFpGroup 47.5 Operations for Finitely Presented Groups 47.5-1 PseudoRandom 47.6 Coset Tables and Coset Enumeration 47.6-1 CosetTable 47.6-2 TracedCosetFpGroup 47.6-3 FactorCosetAction 47.6-4 CosetTableBySubgroup 47.6-5 CosetTableFromGensAndRels 47.6-6 CosetTableDefaultMaxLimit 47.6-7 CosetTableDefaultLimit 47.6-8 MostFrequentGeneratorFpGroup 47.6-9 IndicesInvolutaryGenerators 47.7 Standardization of coset tables 47.7-1 CosetTableStandard 47.7-2 StandardizeTable 47.8 Coset tables for subgroups in the whole group 47.8-1 CosetTableInWholeGroup 47.8-2 SubgroupOfWholeGroupByCosetTable 47.9 Augmented Coset Tables and Rewriting 47.9-1 AugmentedCosetTableInWholeGroup 47.9-2 AugmentedCosetTableMtc 47.9-3 AugmentedCosetTableRrs 47.9-4 RewriteWord 47.10 Low Index Subgroups 47.10-1 LowIndexSubgroupsFpGroupIterator 47.11 Converting Groups to Finitely Presented Groups 47.11-1 IsomorphismFpGroup 47.11-2 IsomorphismFpGroupByGenerators 47.12 New Presentations and Presentations for Subgroups 47.12-1 IsomorphismSimplifiedFpGroup 47.13 Preimages under Homomorphisms from an FpGroup 47.13-1 SubgroupOfWholeGroupByQuotientSubgroup 47.13-2 IsSubgroupOfWholeGroupByQuotientRep 47.13-3 AsSubgroupOfWholeGroupByQuotient 47.13-4 DefiningQuotientHomomorphism 47.14 Quotient Methods 47.14-1 PQuotient 47.14-2 EpimorphismQuotientSystem 47.14-3 EpimorphismPGroup 47.14-4 EpimorphismNilpotentQuotient 47.14-5 SolvableQuotient 47.14-6 EpimorphismSolvableQuotient 47.14-7 LargerQuotientBySubgroupAbelianization 47.15 Abelian Invariants for Subgroups 47.15-1 AbelianInvariantsSubgroupFpGroup 47.15-2 AbelianInvariantsSubgroupFpGroupMtc 47.15-3 AbelianInvariantsSubgroupFpGroupRrs 47.15-4 AbelianInvariantsNormalClosureFpGroup 47.15-5 AbelianInvariantsNormalClosureFpGroupRrs 47.16 Testing Finiteness of Finitely Presented Groups 47.16-1 IsInfiniteAbelianizationGroup 47.16-2 NewmanInfinityCriterion 48 Presentations and Tietze Transformations 48.1 Creating Presentations 48.1-1 PresentationFpGroup 48.1-2 TzSort 48.1-3 GeneratorsOfPresentation 48.1-4 FpGroupPresentation 48.1-5 PresentationViaCosetTable 48.1-6 SimplifiedFpGroup 48.2 Subgroup Presentations 48.2-1 PresentationSubgroup 48.2-2 PresentationSubgroupRrs 48.2-3 PrimaryGeneratorWords 48.2-4 PresentationSubgroupMtc 48.2-5 PresentationNormalClosureRrs 48.2-6 PresentationNormalClosure 48.3 Relators in a Presentation 48.3-1 TietzeWordAbstractWord 48.3-2 AbstractWordTietzeWord 48.4 Printing Presentations 48.4-1 TzPrintGenerators 48.4-2 TzPrintRelators 48.4-3 TzPrintLengths 48.4-4 TzPrintStatus 48.4-5 TzPrintPresentation 48.4-6 TzPrint 48.4-7 TzPrintPairs 48.5 Changing Presentations 48.5-1 AddGenerator 48.5-2 TzNewGenerator 48.5-3 AddRelator 48.5-4 RemoveRelator 48.6 Tietze Transformations 48.6-1 TzGo 48.6-2 SimplifyPresentation 48.6-3 TzGoGo 48.7 Elementary Tietze Transformations 48.7-1 TzEliminate 48.7-2 TzSearch 48.7-3 TzSearchEqual 48.7-4 TzFindCyclicJoins 48.8 Tietze Transformations that introduce new Generators 48.8-1 TzSubstitute 48.8-2 TzSubstituteCyclicJoins 48.9 Tracing generator images through Tietze transformations 48.9-1 TzInitGeneratorImages 48.9-2 OldGeneratorsOfPresentation 48.9-3 TzImagesOldGens 48.9-4 TzPreImagesNewGens 48.9-5 TzPrintGeneratorImages 48.10 The Decoding Tree Procedure 48.10-1 DecodeTree 48.11 Tietze Options 48.11-1 TzOptions 48.11-2 TzPrintOptions 49 Group Products 49.1 Direct Products 49.1-1 DirectProduct 49.2 Semidirect Products 49.2-1 SemidirectProduct 49.3 Subdirect Products 49.3-1 SubdirectProduct 49.3-2 SubdirectProducts 49.4 Wreath Products 49.4-1 WreathProduct 49.4-2 WreathProductImprimitiveAction 49.4-3 WreathProductProductAction 49.4-4 KuKGenerators 49.5 Free Products 49.5-1 FreeProduct 49.6 Embeddings and Projections for Group Products 49.6-1 Embedding 49.6-2 Projection 50 Group Libraries 50.1 Basic Groups 50.1-1 TrivialGroup 50.1-2 CyclicGroup 50.1-3 AbelianGroup 50.1-4 ElementaryAbelianGroup 50.1-5 FreeAbelianGroup 50.1-6 DihedralGroup 50.1-7 QuaternionGroup 50.1-8 ExtraspecialGroup 50.1-9 AlternatingGroup 50.1-10 SymmetricGroup 50.1-11 MathieuGroup 50.1-12 SuzukiGroup 50.1-13 ReeGroup 50.2 Classical Groups 50.2-1 GeneralLinearGroup 50.2-2 SpecialLinearGroup 50.2-3 GeneralUnitaryGroup 50.2-4 SpecialUnitaryGroup 50.2-5 SymplecticGroup 50.2-6 GeneralOrthogonalGroup 50.2-7 SpecialOrthogonalGroup 50.2-8 Omega 50.2-9 GeneralSemilinearGroup 50.2-10 SpecialSemilinearGroup 50.2-11 ProjectiveGeneralLinearGroup 50.2-12 ProjectiveSpecialLinearGroup 50.2-13 ProjectiveGeneralUnitaryGroup 50.2-14 ProjectiveSpecialUnitaryGroup 50.2-15 ProjectiveSymplecticGroup 50.2-16 ProjectiveOmega 50.3 Conjugacy Classes in Classical Groups 50.3-1 NrConjugacyClassesGL 50.4 Constructors for Basic Groups 50.5 Selection Functions 50.6 Transitive Permutation Groups 50.6-1 TransitiveGroup 50.6-2 NrTransitiveGroups 50.6-3 TransitiveIdentification 50.7 Small Groups 50.7-1 SmallGroup 50.7-2 AllSmallGroups 50.7-3 OneSmallGroup 50.7-4 NumberSmallGroups 50.7-5 IdSmallGroup 50.7-6 IdsOfAllSmallGroups 50.7-7 IdGap3SolvableGroup 50.7-8 SmallGroupsInformation 50.7-9 UnloadSmallGroupsData 50.8 Finite Perfect Groups 50.8-1 SizesPerfectGroups 50.8-2 PerfectGroup 50.8-3 PerfectIdentification 50.8-4 NumberPerfectGroups 50.8-5 NumberPerfectLibraryGroups 50.8-6 SizeNumbersPerfectGroups 50.8-7 DisplayInformationPerfectGroups 50.8-8 More about the Perfect Groups Library 50.9 Primitive Permutation Groups 50.9-1 PrimitiveGroup 50.9-2 NrPrimitiveGroups 50.9-3 PrimitiveGroupsIterator 50.9-4 COHORTS_PRIMITIVE_GROUPS 50.10 Index numbers of primitive groups 50.10-1 PrimitiveIdentification 50.10-2 SimsNo 50.10-3 PRIMITIVE_INDICES_MAGMA 50.11 Irreducible Solvable Matrix Groups 50.11-1 IrreducibleSolvableGroupMS 50.11-2 NumberIrreducibleSolvableGroups 50.11-3 AllIrreducibleSolvableGroups 50.11-4 OneIrreducibleSolvableGroup 50.11-5 PrimitiveIndexIrreducibleSolvableGroup 50.11-6 IrreducibleSolvableGroup 50.12 Irreducible Maximal Finite Integral Matrix Groups 50.12-1 ImfNumberQQClasses 50.12-2 DisplayImfInvariants 50.12-3 ImfInvariants 50.12-4 ImfMatrixGroup 50.12-5 IsomorphismPermGroup 50.12-6 IsomorphismPermGroupImfGroup 51 Semigroups and Monoids 51.1 Semigroups 51.1-1 IsSemigroup 51.1-2 Semigroup 51.1-3 Subsemigroup 51.1-4 IsSubsemigroup 51.1-5 SemigroupByGenerators 51.1-6 AsSemigroup 51.1-7 AsSubsemigroup 51.1-8 GeneratorsOfSemigroup 51.1-9 IsGeneratorsOfSemigroup 51.1-10 FreeSemigroup 51.1-11 SemigroupByMultiplicationTable 51.2 Monoids 51.2-1 IsMonoid 51.2-2 Monoid 51.2-3 Submonoid 51.2-4 MonoidByGenerators 51.2-5 AsMonoid 51.2-6 AsSubmonoid 51.2-7 GeneratorsOfMonoid 51.2-8 TrivialSubmonoid 51.2-9 FreeMonoid 51.2-10 MonoidByMultiplicationTable 51.3 Inverse semigroups and monoids 51.3-1 InverseSemigroup 51.3-2 InverseMonoid 51.3-3 GeneratorsOfInverseSemigroup 51.3-4 GeneratorsOfInverseMonoid 51.3-5 IsInverseSubsemigroup 51.4 Properties of Semigroups 51.4-1 IsRegularSemigroup 51.4-2 IsRegularSemigroupElement 51.4-3 InversesOfSemigroupElement 51.4-4 IsSimpleSemigroup 51.4-5 IsZeroSimpleSemigroup 51.4-6 IsZeroGroup 51.4-7 IsReesCongruenceSemigroup 51.4-8 IsInverseSemigroup 51.5 Ideals of semigroups 51.5-1 SemigroupIdealByGenerators 51.5-2 ReesCongruenceOfSemigroupIdeal 51.5-3 IsLeftSemigroupIdeal 51.6 Congruences for semigroups 51.6-1 IsSemigroupCongruence 51.6-2 IsReesCongruence 51.7 Quotients 51.7-1 IsQuotientSemigroup 51.7-2 HomomorphismQuotientSemigroup 51.7-3 QuotientSemigroupPreimage 51.8 Green's Relations 51.8-1 GreensRRelation 51.8-2 IsGreensRelation 51.8-3 IsGreensClass 51.8-4 IsGreensLessThanOrEqual 51.8-5 RClassOfHClass 51.8-6 EggBoxOfDClass 51.8-7 DisplayEggBoxOfDClass 51.8-8 GreensRClassOfElement 51.8-9 GreensRClasses 51.8-10 GroupHClassOfGreensDClass 51.8-11 IsGroupHClass 51.8-12 IsRegularDClass 51.9 Rees Matrix Semigroups 51.9-1 ReesMatrixSemigroup 51.9-2 ReesMatrixSubsemigroup 51.9-3 IsomorphismReesMatrixSemigroup 51.9-4 IsReesMatrixSemigroupElement 51.9-5 ReesMatrixSemigroupElement 51.9-6 IsReesMatrixSubsemigroup 51.9-7 IsReesMatrixSemigroup 51.9-8 Matrix 51.9-9 Rows and columns 51.9-10 UnderlyingSemigroup 51.9-11 AssociatedReesMatrixSemigroupOfDClass 52 Finitely Presented Semigroups and Monoids 52.1 IsSubsemigroupFpSemigroup (Filter) 52.1-1 IsSubsemigroupFpSemigroup 52.1-2 IsSubmonoidFpMonoid 52.1-3 IsFpSemigroup 52.1-4 IsFpMonoid 52.1-5 IsElementOfFpSemigroup 52.1-6 IsElementOfFpMonoid 52.1-7 FpGrpMonSmgOfFpGrpMonSmgElement 52.2 Creating Finitely Presented Semigroups 52.2-1 \/ 52.2-2 FactorFreeSemigroupByRelations 52.2-3 IsomorphismFpSemigroup 52.3 Comparison of Elements of Finitely Presented Semigroups 52.3-1 \= 52.4 Preimages in the Free Semigroup 52.4-1 UnderlyingElement 52.4-2 ElementOfFpSemigroup 52.4-3 FreeSemigroupOfFpSemigroup 52.4-4 FreeGeneratorsOfFpSemigroup 52.4-5 RelationsOfFpSemigroup 52.5 Finitely presented monoids 52.5-1 \/ 52.6 Rewriting Systems and the Knuth-Bendix Procedure 52.6-1 ReducedConfluentRewritingSystem 52.6-2 KB_REW 52.6-3 KnuthBendixRewritingSystem 52.6-4 SemigroupOfRewritingSystem 52.6-5 MonoidOfRewritingSystem 52.6-6 FreeSemigroupOfRewritingSystem 52.6-7 FreeMonoidOfRewritingSystem 52.7 Todd-Coxeter Procedure 52.7-1 CosetTableOfFpSemigroup 53 Transformations 53.1 The family and categories of transformations 53.1-1 IsTransformation 53.1-2 IsTransformationCollection 53.1-3 TransformationFamily 53.2 Creating transformations 53.2-1 Transformation 53.2-2 Transformation 53.2-3 TransformationByImageAndKernel 53.2-4 Idempotent 53.2-5 TransformationOp 53.2-6 TransformationNumber 53.2-7 RandomTransformation 53.2-8 IdentityTransformation 53.2-9 ConstantTransformation 53.3 Changing the representation of a transformation 53.3-1 AsTransformation 53.3-2 RestrictedTransformation 53.3-3 PermutationOfImage 53.4 Operators for transformations 53.4-1 PermLeftQuoTransformation 53.4-2 IsInjectiveListTrans 53.4-3 ComponentTransformationInt 53.4-4 PreImagesOfTransformation 53.5 Attributes for transformations 53.5-1 DegreeOfTransformation 53.5-2 ImageListOfTransformation 53.5-3 ImageSetOfTransformation 53.5-4 RankOfTransformation 53.5-5 MovedPoints 53.5-6 NrMovedPoints 53.5-7 SmallestMovedPoint 53.5-8 LargestMovedPoint 53.5-9 SmallestImageOfMovedPoint 53.5-10 LargestImageOfMovedPoint 53.5-11 FlatKernelOfTransformation 53.5-12 KernelOfTransformation 53.5-13 InverseOfTransformation 53.5-14 Inverse 53.5-15 IndexPeriodOfTransformation 53.5-16 SmallestIdempotentPower 53.5-17 ComponentsOfTransformation 53.5-18 NrComponentsOfTransformation 53.5-19 ComponentRepsOfTransformation 53.5-20 CyclesOfTransformation 53.5-21 CycleTransformationInt 53.5-22 LeftOne 53.5-23 TrimTransformation 53.6 Displaying transformations 53.7 Semigroups of transformations 53.7-1 IsTransformationSemigroup 53.7-2 DegreeOfTransformationSemigroup 53.7-3 FullTransformationSemigroup 53.7-4 IsFullTransformationSemigroup 53.7-5 IsomorphismTransformationSemigroup 53.7-6 AntiIsomorphismTransformationSemigroup 54 Partial permutations 54.1 The family and categories of partial permutations 54.1-1 IsPartialPerm 54.1-2 IsPartialPermCollection 54.1-3 PartialPermFamily 54.2 Creating partial permutations 54.2-1 PartialPerm 54.2-2 PartialPermOp 54.2-3 RestrictedPartialPerm 54.2-4 JoinOfPartialPerms 54.2-5 MeetOfPartialPerms 54.2-6 EmptyPartialPerm 54.2-7 RandomPartialPerm 54.3 Attributes for partial permutations 54.3-1 DegreeOfPartialPerm 54.3-2 CodegreeOfPartialPerm 54.3-3 RankOfPartialPerm 54.3-4 DomainOfPartialPerm 54.3-5 ImageOfPartialPermCollection 54.3-6 ImageListOfPartialPerm 54.3-7 ImageSetOfPartialPerm 54.3-8 FixedPointsOfPartialPerm 54.3-9 MovedPoints 54.3-10 NrFixedPoints 54.3-11 NrMovedPoints 54.3-12 SmallestMovedPoint 54.3-13 LargestMovedPoint 54.3-14 SmallestImageOfMovedPoint 54.3-15 LargestImageOfMovedPoint 54.3-16 IndexPeriodOfPartialPerm 54.3-17 SmallestIdempotentPower 54.3-18 ComponentsOfPartialPerm 54.3-19 NrComponentsOfPartialPerm 54.3-20 ComponentRepsOfPartialPerm 54.3-21 LeftOne 54.3-22 One 54.3-23 Zero 54.4 Changing the representation of a partial permutation 54.4-1 AsPartialPerm 54.4-2 AsPartialPerm 54.5 Operators and operations for partial permutations 54.5-1 PermLeftQuoPartialPerm 54.5-2 PreImagePartialPerm 54.5-3 ComponentPartialPermInt 54.5-4 NaturalLeqPartialPerm 54.5-5 ShortLexLeqPartialPerm 54.5-6 TrimPartialPerm 54.6 Displaying partial permutations 54.7 Semigroups and inverse semigroups of partial permutations 54.7-1 IsPartialPermSemigroup 54.7-2 DegreeOfPartialPermSemigroup 54.7-3 SymmetricInverseSemigroup 54.7-4 IsSymmetricInverseSemigroup 54.7-5 NaturalPartialOrder 54.7-6 IsomorphismPartialPermMonoid 55 Additive Magmas 55.1 (Near-)Additive Magma Categories 55.1-1 IsNearAdditiveMagma 55.1-2 IsNearAdditiveMagmaWithZero 55.1-3 IsNearAdditiveGroup 55.1-4 IsAdditiveMagma 55.1-5 IsAdditiveMagmaWithZero 55.1-6 IsAdditiveGroup 55.2 (Near-)Additive Magma Generation 55.2-1 NearAdditiveMagma 55.2-2 NearAdditiveMagmaWithZero 55.2-3 NearAdditiveGroup 55.2-4 NearAdditiveMagmaByGenerators 55.2-5 NearAdditiveMagmaWithZeroByGenerators 55.2-6 NearAdditiveGroupByGenerators 55.2-7 SubnearAdditiveMagma 55.2-8 SubnearAdditiveMagmaWithZero 55.2-9 SubnearAdditiveGroup 55.3 Attributes and Properties for (Near-)Additive Magmas 55.3-1 IsAdditivelyCommutative 55.3-2 GeneratorsOfNearAdditiveMagma 55.3-3 GeneratorsOfNearAdditiveMagmaWithZero 55.3-4 GeneratorsOfNearAdditiveGroup 55.3-5 AdditiveNeutralElement 55.3-6 TrivialSubnearAdditiveMagmaWithZero 55.4 Operations for (Near-)Additive Magmas 55.4-1 ClosureNearAdditiveGroup 55.4-2 ShowAdditionTable 56 Rings 56.1 Generating Rings 56.1-1 IsRing 56.1-2 Ring 56.1-3 DefaultRing 56.1-4 RingByGenerators 56.1-5 DefaultRingByGenerators 56.1-6 GeneratorsOfRing 56.1-7 Subring 56.1-8 ClosureRing 56.1-9 Quotient 56.2 Ideals of Rings 56.2-1 TwoSidedIdeal 56.2-2 TwoSidedIdealNC 56.2-3 IsTwoSidedIdeal 56.2-4 TwoSidedIdealByGenerators 56.2-5 LeftIdealByGenerators 56.2-6 RightIdealByGenerators 56.2-7 GeneratorsOfTwoSidedIdeal 56.2-8 GeneratorsOfLeftIdeal 56.2-9 GeneratorsOfRightIdeal 56.2-10 LeftActingRingOfIdeal 56.2-11 AsLeftIdeal 56.3 Rings With One 56.3-1 IsRingWithOne 56.3-2 RingWithOne 56.3-3 RingWithOneByGenerators 56.3-4 GeneratorsOfRingWithOne 56.3-5 SubringWithOne 56.4 Properties of Rings 56.4-1 IsIntegralRing 56.4-2 IsUniqueFactorizationRing 56.4-3 IsLDistributive 56.4-4 IsRDistributive 56.4-5 IsDistributive 56.4-6 IsAnticommutative 56.4-7 IsZeroSquaredRing 56.4-8 IsJacobianRing 56.5 Units and Factorizations 56.5-1 IsUnit 56.5-2 Units 56.5-3 IsAssociated 56.5-4 Associates 56.5-5 StandardAssociate 56.5-6 StandardAssociateUnit 56.5-7 IsIrreducibleRingElement 56.5-8 IsPrime 56.5-9 Factors 56.5-10 PadicValuation 56.6 Euclidean Rings 56.6-1 IsEuclideanRing 56.6-2 EuclideanDegree 56.6-3 EuclideanQuotient 56.6-4 EuclideanRemainder 56.6-5 QuotientRemainder 56.7 Gcd and Lcm 56.7-1 Gcd 56.7-2 GcdOp 56.7-3 GcdRepresentation 56.7-4 GcdRepresentationOp 56.7-5 ShowGcd 56.7-6 Lcm 56.7-7 LcmOp 56.7-8 QuotientMod 56.7-9 PowerMod 56.7-10 InterpolatedPolynomial 56.8 Homomorphisms of Rings 56.8-1 RingGeneralMappingByImages 56.8-2 RingHomomorphismByImages 56.8-3 RingHomomorphismByImagesNC 56.8-4 NaturalHomomorphismByIdeal 56.9 Small Rings 56.9-1 SmallRing 56.9-2 NumberSmallRings 56.9-3 Subrings 56.9-4 Ideals 56.9-5 DirectSum 56.9-6 RingByStructureConstants 57 Modules 57.1 Generating modules 57.1-1 IsLeftOperatorAdditiveGroup 57.1-2 IsLeftModule 57.1-3 GeneratorsOfLeftOperatorAdditiveGroup 57.1-4 GeneratorsOfLeftModule 57.1-5 AsLeftModule 57.1-6 IsRightOperatorAdditiveGroup 57.1-7 IsRightModule 57.1-8 GeneratorsOfRightOperatorAdditiveGroup 57.1-9 GeneratorsOfRightModule 57.1-10 LeftModuleByGenerators 57.1-11 LeftActingDomain 57.2 Submodules 57.2-1 Submodule 57.2-2 SubmoduleNC 57.2-3 ClosureLeftModule 57.2-4 TrivialSubmodule 57.3 Free Modules 57.3-1 IsFreeLeftModule 57.3-2 FreeLeftModule 57.3-3 Dimension 57.3-4 IsFiniteDimensional 57.3-5 UseBasis 57.3-6 IsRowModule 57.3-7 IsMatrixModule 57.3-8 IsFullRowModule 57.3-9 FullRowModule 57.3-10 IsFullMatrixModule 57.3-11 FullMatrixModule 58 Fields and Division Rings 58.1 Generating Fields 58.1-1 IsDivisionRing 58.1-2 IsField 58.1-3 Field 58.1-4 DefaultField 58.1-5 DefaultFieldByGenerators 58.1-6 GeneratorsOfDivisionRing 58.1-7 GeneratorsOfField 58.1-8 DivisionRingByGenerators 58.1-9 AsDivisionRing 58.2 Subfields of Fields 58.2-1 Subfield 58.2-2 FieldOverItselfByGenerators 58.2-3 PrimitiveElement 58.2-4 PrimeField 58.2-5 IsPrimeField 58.2-6 DegreeOverPrimeField 58.2-7 DefiningPolynomial 58.2-8 RootOfDefiningPolynomial 58.2-9 FieldExtension 58.2-10 Subfields 58.3 Galois Action 58.3-1 GaloisGroup 58.3-2 MinimalPolynomial 58.3-3 TracePolynomial 58.3-4 Norm 58.3-5 Traces of field elements and matrices 58.3-6 Conjugates 58.3-7 NormalBase 59 Finite Fields 59.1 Finite Field Elements 59.1-1 IsFFE 59.1-2 Z 59.1-3 IsLexOrderedFFE 59.2 Operations for Finite Field Elements 59.2-1 DegreeFFE 59.2-2 LogFFE 59.2-3 IntFFE 59.2-4 IntFFESymm 59.2-5 IntVecFFE 59.2-6 AsInternalFFE 59.3 Creating Finite Fields 59.3-1 DefaultField 59.3-2 GaloisField 59.3-3 PrimitiveRoot 59.4 Frobenius Automorphisms 59.4-1 FrobeniusAutomorphism 59.5 Conway Polynomials 59.5-1 ConwayPolynomial 59.5-2 IsCheapConwayPolynomial 59.5-3 RandomPrimitivePolynomial 59.6 Printing, Viewing and Displaying Finite Field Elements 59.6-1 ViewObj 60 Abelian Number Fields 60.1 Construction of Abelian Number Fields 60.1-1 CyclotomicField 60.1-2 AbelianNumberField 60.1-3 GaussianRationals 60.2 Operations for Abelian Number Fields 60.2-1 Factors 60.2-2 IsNumberField 60.2-3 IsAbelianNumberField 60.2-4 IsCyclotomicField 60.2-5 GaloisStabilizer 60.3 Integral Bases of Abelian Number Fields 60.3-1 ZumbroichBase 60.3-2 LenstraBase 60.4 Galois Groups of Abelian Number Fields 60.4-1 GaloisGroup 60.4-2 ANFAutomorphism 60.5 Gaussians 60.5-1 GaussianIntegers 60.5-2 IsGaussianIntegers 61 Vector Spaces 61.1 IsLeftVectorSpace (Filter) 61.1-1 IsLeftVectorSpace 61.2 Constructing Vector Spaces 61.2-1 VectorSpace 61.2-2 Subspace 61.2-3 AsVectorSpace 61.2-4 AsSubspace 61.3 Operations and Attributes for Vector Spaces 61.3-1 GeneratorsOfLeftVectorSpace 61.3-2 TrivialSubspace 61.4 Domains of Subspaces of Vector Spaces 61.4-1 Subspaces 61.4-2 IsSubspacesVectorSpace 61.5 Bases of Vector Spaces 61.5-1 IsBasis 61.5-2 Basis 61.5-3 CanonicalBasis 61.5-4 RelativeBasis 61.6 Operations for Vector Space Bases 61.6-1 BasisVectors 61.6-2 UnderlyingLeftModule 61.6-3 Coefficients 61.6-4 LinearCombination 61.6-5 EnumeratorByBasis 61.6-6 IteratorByBasis 61.7 Operations for Special Kinds of Bases 61.7-1 IsCanonicalBasis 61.7-2 IsIntegralBasis 61.7-3 IsNormalBasis 61.8 Mutable Bases 61.8-1 IsMutableBasis 61.8-2 MutableBasis 61.8-3 NrBasisVectors 61.8-4 ImmutableBasis 61.8-5 IsContainedInSpan 61.8-6 CloseMutableBasis 61.9 Row and Matrix Spaces 61.9-1 IsRowSpace 61.9-2 IsMatrixSpace 61.9-3 IsGaussianSpace 61.9-4 FullRowSpace 61.9-5 FullMatrixSpace 61.9-6 DimensionOfVectors 61.9-7 IsSemiEchelonized 61.9-8 SemiEchelonBasis 61.9-9 IsCanonicalBasisFullRowModule 61.9-10 IsCanonicalBasisFullMatrixModule 61.9-11 NormedRowVectors 61.9-12 SiftedVector 61.10 Vector Space Homomorphisms 61.10-1 LeftModuleGeneralMappingByImages 61.10-2 LeftModuleHomomorphismByImages 61.10-3 LeftModuleHomomorphismByMatrix 61.10-4 NaturalHomomorphismBySubspace 61.10-5 Hom 61.10-6 End 61.10-7 IsFullHomModule 61.10-8 IsPseudoCanonicalBasisFullHomModule 61.10-9 IsLinearMappingsModule 61.11 Vector Spaces Handled By Nice Bases 61.11-1 NiceFreeLeftModule 61.11-2 NiceVector 61.11-3 NiceFreeLeftModuleInfo 61.11-4 NiceBasis 61.11-5 IsBasisByNiceBasis 61.11-6 IsHandledByNiceBasis 61.12 How to Implement New Kinds of Vector Spaces 61.12-1 DeclareHandlingByNiceBasis 61.12-2 NiceBasisFiltersInfo 61.12-3 CheckForHandlingByNiceBasis 62 Algebras 62.1 InfoAlgebra (Info Class) 62.1-1 InfoAlgebra 62.2 Constructing Algebras by Generators 62.2-1 Algebra 62.2-2 AlgebraWithOne 62.3 Constructing Algebras as Free Algebras 62.3-1 FreeAlgebra 62.3-2 FreeAlgebraWithOne 62.3-3 FreeAssociativeAlgebra 62.3-4 FreeAssociativeAlgebraWithOne 62.4 Constructing Algebras by Structure Constants 62.4-1 AlgebraByStructureConstants 62.4-2 StructureConstantsTable 62.4-3 EmptySCTable 62.4-4 SetEntrySCTable 62.4-5 GapInputSCTable 62.4-6 TestJacobi 62.4-7 IdentityFromSCTable 62.4-8 QuotientFromSCTable 62.5 Some Special Algebras 62.5-1 QuaternionAlgebra 62.5-2 ComplexificationQuat 62.5-3 OctaveAlgebra 62.5-4 FullMatrixAlgebra 62.5-5 NullAlgebra 62.6 Subalgebras 62.6-1 Subalgebra 62.6-2 SubalgebraNC 62.6-3 SubalgebraWithOne 62.6-4 SubalgebraWithOneNC 62.6-5 TrivialSubalgebra 62.7 Ideals of Algebras 62.8 Categories and Properties of Algebras 62.8-1 IsFLMLOR 62.8-2 IsFLMLORWithOne 62.8-3 IsAlgebra 62.8-4 IsAlgebraWithOne 62.8-5 IsLieAlgebra 62.8-6 IsSimpleAlgebra 62.8-7 IsFiniteDimensional 62.8-8 IsQuaternion 62.9 Attributes and Operations for Algebras 62.9-1 GeneratorsOfAlgebra 62.9-2 GeneratorsOfAlgebraWithOne 62.9-3 ProductSpace 62.9-4 PowerSubalgebraSeries 62.9-5 AdjointBasis 62.9-6 IndicesOfAdjointBasis 62.9-7 AsAlgebra 62.9-8 AsAlgebraWithOne 62.9-9 AsSubalgebra 62.9-10 AsSubalgebraWithOne 62.9-11 MutableBasisOfClosureUnderAction 62.9-12 MutableBasisOfNonassociativeAlgebra 62.9-13 MutableBasisOfIdealInNonassociativeAlgebra 62.9-14 DirectSumOfAlgebras 62.9-15 FullMatrixAlgebraCentralizer 62.9-16 RadicalOfAlgebra 62.9-17 CentralIdempotentsOfAlgebra 62.9-18 DirectSumDecomposition 62.9-19 LeviMalcevDecomposition 62.9-20 Grading 62.10 Homomorphisms of Algebras 62.10-1 AlgebraGeneralMappingByImages 62.10-2 AlgebraHomomorphismByImages 62.10-3 AlgebraHomomorphismByImagesNC 62.10-4 AlgebraWithOneGeneralMappingByImages 62.10-5 AlgebraWithOneHomomorphismByImages 62.10-6 AlgebraWithOneHomomorphismByImagesNC 62.10-7 NaturalHomomorphismByIdeal 62.10-8 OperationAlgebraHomomorphism 62.10-9 NiceAlgebraMonomorphism 62.10-10 IsomorphismFpAlgebra 62.10-11 IsomorphismMatrixAlgebra 62.10-12 IsomorphismSCAlgebra 62.10-13 RepresentativeLinearOperation 62.11 Representations of Algebras 62.11-1 LeftAlgebraModuleByGenerators 62.11-2 RightAlgebraModuleByGenerators 62.11-3 BiAlgebraModuleByGenerators 62.11-4 LeftAlgebraModule 62.11-5 RightAlgebraModule 62.11-6 BiAlgebraModule 62.11-7 GeneratorsOfAlgebraModule 62.11-8 IsAlgebraModuleElement 62.11-9 IsLeftAlgebraModuleElement 62.11-10 IsRightAlgebraModuleElement 62.11-11 LeftActingAlgebra 62.11-12 RightActingAlgebra 62.11-13 ActingAlgebra 62.11-14 IsBasisOfAlgebraModuleElementSpace 62.11-15 MatrixOfAction 62.11-16 SubAlgebraModule 62.11-17 LeftModuleByHomomorphismToMatAlg 62.11-18 RightModuleByHomomorphismToMatAlg 62.11-19 AdjointModule 62.11-20 FaithfulModule 62.11-21 ModuleByRestriction 62.11-22 NaturalHomomorphismBySubAlgebraModule 62.11-23 DirectSumOfAlgebraModules 62.11-24 TranslatorSubalgebra 63 Finitely Presented Algebras 64 Lie Algebras 64.1 Lie Objects 64.1-1 LieObject 64.1-2 IsLieObject 64.1-3 LieFamily 64.1-4 UnderlyingFamily 64.1-5 UnderlyingRingElement 64.2 Constructing Lie algebras 64.2-1 LieAlgebraByStructureConstants 64.2-2 RestrictedLieAlgebraByStructureConstants 64.2-3 LieAlgebra 64.2-4 FreeLieAlgebra 64.2-5 FullMatrixLieAlgebra 64.2-6 RightDerivations 64.2-7 SimpleLieAlgebra 64.3 Distinguished Subalgebras 64.3-1 LieCentre 64.3-2 LieCentralizer 64.3-3 LieNormalizer 64.3-4 LieDerivedSubalgebra 64.3-5 LieNilRadical 64.3-6 LieSolvableRadical 64.3-7 CartanSubalgebra 64.4 Series of Ideals 64.4-1 LieDerivedSeries 64.4-2 LieLowerCentralSeries 64.4-3 LieUpperCentralSeries 64.5 Properties of a Lie Algebra 64.5-1 IsLieAbelian 64.5-2 IsLieNilpotent 64.5-3 IsLieSolvable 64.6 Semisimple Lie Algebras and Root Systems 64.6-1 SemiSimpleType 64.6-2 ChevalleyBasis 64.6-3 IsRootSystem 64.6-4 IsRootSystemFromLieAlgebra 64.6-5 RootSystem 64.6-6 UnderlyingLieAlgebra 64.6-7 PositiveRoots 64.6-8 NegativeRoots 64.6-9 PositiveRootVectors 64.6-10 NegativeRootVectors 64.6-11 SimpleSystem 64.6-12 CartanMatrix 64.6-13 BilinearFormMat 64.6-14 CanonicalGenerators 64.7 Semisimple Lie Algebras and Weyl Groups of Root Systems 64.7-1 IsWeylGroup 64.7-2 SparseCartanMatrix 64.7-3 WeylGroup 64.7-4 ApplySimpleReflection 64.7-5 LongestWeylWordPerm 64.7-6 ConjugateDominantWeight 64.7-7 WeylOrbitIterator 64.8 Restricted Lie algebras 64.8-1 IsRestrictedLieAlgebra 64.8-2 PthPowerImages 64.8-3 PthPowerImage 64.8-4 JenningsLieAlgebra 64.8-5 PCentralLieAlgebra 64.8-6 NaturalHomomorphismOfLieAlgebraFromNilpotentGroup 64.9 The Adjoint Representation 64.9-1 AdjointMatrix 64.9-2 AdjointAssociativeAlgebra 64.9-3 KillingMatrix 64.9-4 KappaPerp 64.9-5 IsNilpotentElement 64.9-6 NonNilpotentElement 64.9-7 FindSl2 64.10 Universal Enveloping Algebras 64.10-1 UniversalEnvelopingAlgebra 64.11 Finitely Presented Lie Algebras 64.11-1 FpLieAlgebraByCartanMatrix 64.11-2 NilpotentQuotientOfFpLieAlgebra 64.12 Modules over Lie Algebras and Their Cohomology 64.12-1 IsCochain 64.12-2 Cochain 64.12-3 CochainSpace 64.12-4 ValueCochain 64.12-5 LieCoboundaryOperator 64.12-6 Cocycles 64.12-7 Coboundaries 64.13 Modules over Semisimple Lie Algebras 64.13-1 DominantWeights 64.13-2 DominantCharacter 64.13-3 DecomposeTensorProduct 64.13-4 DimensionOfHighestWeightModule 64.14 Admissible Lattices in UEA 64.14-1 IsUEALatticeElement 64.14-2 LatticeGeneratorsInUEA 64.14-3 ObjByExtRep 64.14-4 IsWeightRepElement 64.14-5 HighestWeightModule 64.15 Tensor Products and Exterior and Symmetric Powers 64.15-1 TensorProductOfAlgebraModules 64.15-2 ExteriorPowerOfAlgebraModule 64.15-3 SymmetricPowerOfAlgebraModule 65 Magma Rings 65.1 Free Magma Rings 65.1-1 FreeMagmaRing 65.1-2 GroupRing 65.1-3 IsFreeMagmaRing 65.1-4 IsFreeMagmaRingWithOne 65.1-5 IsGroupRing 65.1-6 UnderlyingMagma 65.1-7 AugmentationIdeal 65.2 Elements of Free Magma Rings 65.2-1 IsMagmaRingObjDefaultRep 65.2-2 IsElementOfFreeMagmaRing 65.2-3 IsElementOfFreeMagmaRingFamily 65.2-4 CoefficientsAndMagmaElements 65.2-5 ZeroCoefficient 65.2-6 ElementOfMagmaRing 65.3 Natural Embeddings related to Magma Rings 65.4 Magma Rings modulo Relations 65.4-1 IsElementOfMagmaRingModuloRelations 65.4-2 IsElementOfMagmaRingModuloRelationsFamily 65.4-3 NormalizedElementOfMagmaRingModuloRelations 65.4-4 IsMagmaRingModuloRelations 65.5 Magma Rings modulo the Span of a Zero Element 65.5-1 IsElementOfMagmaRingModuloSpanOfZeroFamily 65.5-2 IsMagmaRingModuloSpanOfZero 65.5-3 MagmaRingModuloSpanOfZero 65.6 Technical Details about the Implementation of Magma Rings 66 Polynomials and Rational Functions 66.1 Indeterminates 66.1-1 Indeterminate 66.1-2 IndeterminateNumberOfUnivariateRationalFunction 66.1-3 IndeterminateOfUnivariateRationalFunction 66.1-4 IndeterminateName 66.1-5 CIUnivPols 66.2 Operations for Rational Functions 66.3 Comparison of Rational Functions 66.4 Properties and Attributes of Rational Functions 66.4-1 IsPolynomialFunction 66.4-2 NumeratorOfRationalFunction 66.4-3 DenominatorOfRationalFunction 66.4-4 IsPolynomial 66.4-5 AsPolynomial 66.4-6 IsUnivariateRationalFunction 66.4-7 CoefficientsOfUnivariateRationalFunction 66.4-8 IsUnivariatePolynomial 66.4-9 CoefficientsOfUnivariatePolynomial 66.4-10 IsLaurentPolynomial 66.4-11 IsConstantRationalFunction 66.4-12 IsPrimitivePolynomial 66.4-13 SplittingField 66.5 Univariate Polynomials 66.5-1 UnivariatePolynomial 66.5-2 UnivariatePolynomialByCoefficients 66.5-3 DegreeOfLaurentPolynomial 66.5-4 RootsOfPolynomial 66.5-5 RootsOfUPol 66.5-6 QuotRemLaurpols 66.5-7 UnivariatenessTestRationalFunction 66.5-8 InfoPoly 66.6 Polynomials as Univariate Polynomials in one Indeterminate 66.6-1 DegreeIndeterminate 66.6-2 PolynomialCoefficientsOfPolynomial 66.6-3 LeadingCoefficient 66.6-4 LeadingMonomial 66.6-5 Derivative 66.6-6 Discriminant 66.6-7 Resultant 66.7 Multivariate Polynomials 66.7-1 Value 66.8 Minimal Polynomials 66.8-1 MinimalPolynomial 66.9 Cyclotomic Polynomials 66.9-1 CyclotomicPolynomial 66.10 Polynomial Factorization 66.10-1 Factors 66.10-2 FactorsSquarefree 66.11 Polynomials over the Rationals 66.11-1 PrimitivePolynomial 66.11-2 PolynomialModP 66.11-3 GaloisType 66.11-4 ProbabilityShapes 66.12 Factorization of Polynomials over the Rationals 66.12-1 BombieriNorm 66.12-2 MinimizedBombieriNorm 66.12-3 HenselBound 66.12-4 OneFactorBound 66.13 Laurent Polynomials 66.13-1 LaurentPolynomialByCoefficients 66.13-2 CoefficientsOfLaurentPolynomial 66.13-3 IndeterminateNumberOfLaurentPolynomial 66.14 Univariate Rational Functions 66.14-1 UnivariateRationalFunctionByCoefficients 66.15 Polynomial Rings and Function Fields 66.15-1 PolynomialRing 66.15-2 IndeterminatesOfPolynomialRing 66.15-3 CoefficientsRing 66.15-4 IsPolynomialRing 66.15-5 IsFiniteFieldPolynomialRing 66.15-6 IsAbelianNumberFieldPolynomialRing 66.15-7 IsRationalsPolynomialRing 66.15-8 FunctionField 66.15-9 IsFunctionField 66.16 Univariate Polynomial Rings 66.16-1 UnivariatePolynomialRing 66.16-2 IsUnivariatePolynomialRing 66.17 Monomial Orderings 66.17-1 IsMonomialOrdering 66.17-2 LeadingMonomialOfPolynomial 66.17-3 LeadingTermOfPolynomial 66.17-4 LeadingCoefficientOfPolynomial 66.17-5 MonomialComparisonFunction 66.17-6 MonomialExtrepComparisonFun 66.17-7 MonomialLexOrdering 66.17-8 MonomialGrlexOrdering 66.17-9 MonomialGrevlexOrdering 66.17-10 EliminationOrdering 66.17-11 PolynomialReduction 66.17-12 PolynomialReducedRemainder 66.17-13 PolynomialDivisionAlgorithm 66.17-14 MonomialExtGrlexLess 66.18 Groebner Bases 66.18-1 GroebnerBasis 66.18-2 ReducedGroebnerBasis 66.18-3 StoredGroebnerBasis 66.18-4 InfoGroebner 66.19 Rational Function Families 66.19-1 RationalFunctionsFamily 66.19-2 IsPolynomialFunctionsFamily 66.19-3 CoefficientsFamily 66.20 The Representations of Rational Functions 66.21 The Defining Attributes of Rational Functions 66.21-1 IsRationalFunctionDefaultRep 66.21-2 ExtRepNumeratorRatFun 66.21-3 ExtRepDenominatorRatFun 66.21-4 ZeroCoefficientRatFun 66.21-5 IsPolynomialDefaultRep 66.21-6 ExtRepPolynomialRatFun 66.21-7 IsLaurentPolynomialDefaultRep 66.22 Creation of Rational Functions 66.22-1 RationalFunctionByExtRep 66.22-2 PolynomialByExtRep 66.22-3 LaurentPolynomialByExtRep 66.23 Arithmetic for External Representations of Polynomials 66.23-1 ZippedSum 66.23-2 ZippedProduct 66.23-3 QuotientPolynomialsExtRep 66.24 Cancellation Tests for Rational Functions 66.24-1 RationalFunctionByExtRepWithCancellation 66.24-2 TryGcdCancelExtRepPolynomials 66.24-3 HeuristicCancelPolynomialsExtRep 67 Algebraic extensions of fields 67.1 Creation of Algebraic Extensions 67.1-1 AlgebraicExtension 67.1-2 IsAlgebraicExtension 67.2 Elements in Algebraic Extensions 67.2-1 IsAlgebraicElement 68 p-adic Numbers (preliminary) 68.1 Pure p-adic Numbers 68.1-1 PurePadicNumberFamily 68.1-2 PadicNumber 68.1-3 Valuation 68.1-4 ShiftedPadicNumber 68.1-5 IsPurePadicNumber 68.1-6 IsPurePadicNumberFamily 68.2 Extensions of the p-adic Numbers 68.2-1 PadicExtensionNumberFamily 68.2-2 PadicNumber 68.2-3 IsPadicExtensionNumber 68.2-4 IsPadicExtensionNumberFamily 69 The MeatAxe 69.1 MeatAxe Modules 69.1-1 GModuleByMats 69.2 Module Constructions 69.2-1 PermutationGModule 69.2-2 TensorProductGModule 69.2-3 WedgeGModule 69.3 Selecting a Different MeatAxe 69.3-1 MTX 69.4 Accessing a Module 69.4-1 MTX.Generators 69.4-2 MTX.Dimension 69.4-3 MTX.Field 69.5 Irreducibility Tests 69.5-1 MTX.IsIrreducible 69.5-2 MTX.IsAbsolutelyIrreducible 69.5-3 MTX.DegreeSplittingField 69.6 Decomposition of modules 69.6-1 MTX.IsIndecomposable 69.6-2 MTX.Indecomposition 69.6-3 MTX.HomogeneousComponents 69.7 Finding Submodules 69.7-1 MTX.SubmoduleGModule 69.7-2 MTX.ProperSubmoduleBasis 69.7-3 MTX.BasesSubmodules 69.7-4 MTX.BasesMinimalSubmodules 69.7-5 MTX.BasesMaximalSubmodules 69.7-6 MTX.BasisRadical 69.7-7 MTX.BasisSocle 69.7-8 MTX.BasesMinimalSupermodules 69.7-9 MTX.BasesCompositionSeries 69.7-10 MTX.CompositionFactors 69.7-11 MTX.CollectedFactors 69.8 Induced Actions 69.8-1 MTX.NormedBasisAndBaseChange 69.8-2 MTX.InducedActionSubmodule 69.8-3 MTX.InducedActionFactorModule 69.8-4 MTX.InducedActionMatrix 69.8-5 MTX.InducedAction 69.9 Module Homomorphisms 69.9-1 MTX.BasisModuleHomomorphisms 69.9-2 MTX.BasisModuleEndomorphisms 69.9-3 MTX.IsomorphismModules 69.9-4 MTX.ModuleAutomorphisms 69.10 Module Homomorphisms for irreducible modules 69.10-1 MTX.IsEquivalent 69.10-2 MTX.IsomorphismIrred 69.10-3 MTX.Homomorphism 69.10-4 MTX.Homomorphisms 69.10-5 MTX.Distinguish 69.11 MeatAxe Functionality for Invariant Forms 69.11-1 MTX.InvariantBilinearForm 69.11-2 MTX.InvariantSesquilinearForm 69.11-3 MTX.InvariantQuadraticForm 69.11-4 MTX.BasisInOrbit 69.11-5 MTX.OrthogonalSign 69.12 The Smash MeatAxe 69.12-1 SMTX.RandomIrreducibleSubGModule 69.12-2 SMTX.GoodElementGModule 69.12-3 SMTX.SortHomGModule 69.12-4 SMTX.MinimalSubGModules 69.12-5 SMTX.Setter 69.12-6 SMTX.Getter 69.12-7 SMTX.IrreducibilityTest 69.12-8 SMTX.AbsoluteIrreducibilityTest 69.12-9 SMTX.MinimalSubGModule 69.12-10 SMTX.MatrixSum 69.12-11 SMTX.CompleteBasis 69.13 Smash MeatAxe Flags 69.13-1 SMTX.Subbasis 69.13-2 SMTX.AlgEl 69.13-3 SMTX.AlgElMat 69.13-4 SMTX.AlgElCharPol 69.13-5 SMTX.AlgElCharPolFac 69.13-6 SMTX.AlgElNullspaceVec 69.13-7 SMTX.AlgElNullspaceDimension 69.13-8 SMTX.CentMat 69.13-9 SMTX.CentMatMinPoly 70 Tables of Marks 70.1 More about Tables of Marks 70.2 Table of Marks Objects in GAP 70.3 Constructing Tables of Marks 70.3-1 TableOfMarks 70.3-2 TableOfMarksByLattice 70.3-3 LatticeSubgroupsByTom 70.4 Printing Tables of Marks 70.4-1 ViewObj 70.4-2 PrintObj 70.4-3 Display 70.5 Sorting Tables of Marks 70.5-1 SortedTom 70.5-2 PermutationTom 70.6 Technical Details about Tables of Marks 70.6-1 InfoTom 70.6-2 IsTableOfMarks 70.6-3 TableOfMarksFamily 70.6-4 TableOfMarksComponents 70.6-5 ConvertToTableOfMarks 70.7 Attributes of Tables of Marks 70.7-1 MarksTom 70.7-2 NrSubsTom 70.7-3 LengthsTom 70.7-4 ClassTypesTom 70.7-5 ClassNamesTom 70.7-6 FusionsTom 70.7-7 UnderlyingGroup 70.7-8 IdempotentsTom 70.7-9 Identifier 70.7-10 MatTom 70.7-11 MoebiusTom 70.7-12 WeightsTom 70.8 Properties of Tables of Marks 70.8-1 IsAbelianTom 70.9 Other Operations for Tables of Marks 70.9-1 IsInternallyConsistent 70.9-2 DerivedSubgroupTom 70.9-3 DerivedSubgroupsTomPossible 70.9-4 NormalizerTom 70.9-5 ContainedTom 70.9-6 ContainingTom 70.9-7 CyclicExtensionsTom 70.9-8 DecomposedFixedPointVector 70.9-9 EulerianFunctionByTom 70.9-10 IntersectionsTom 70.9-11 FactorGroupTom 70.9-12 MaximalSubgroupsTom 70.9-13 MinimalSupergroupsTom 70.10 Accessing Subgroups via Tables of Marks 70.10-1 GeneratorsSubgroupsTom 70.10-2 StraightLineProgramsTom 70.10-3 IsTableOfMarksWithGens 70.10-4 RepresentativeTom 70.11 The Interface between Tables of Marks and Character Tables 70.11-1 FusionCharTableTom 70.11-2 PermCharsTom 70.12 Generic Construction of Tables of Marks 70.12-1 TableOfMarksCyclic 70.12-2 TableOfMarksDihedral 70.12-3 TableOfMarksFrobenius 70.13 The Library of Tables of Marks 71 Character Tables 71.1 Some Remarks about Character Theory in GAP 71.2 History of Character Theory Stuff in GAP 71.3 Creating Character Tables 71.3-1 CharacterTable 71.3-2 BrauerTable 71.3-3 CharacterTableRegular 71.3-4 SupportedCharacterTableInfo 71.3-5 ConvertToCharacterTable 71.4 Character Table Categories 71.4-1 IsNearlyCharacterTable 71.4-2 InfoCharacterTable 71.4-3 NearlyCharacterTablesFamily 71.5 Conventions for Character Tables 71.6 The Interface between Character Tables and Groups 71.6-1 UnderlyingGroup 71.6-2 ConjugacyClasses 71.6-3 IdentificationOfConjugacyClasses 71.6-4 CharacterTableWithStoredGroup 71.6-5 CompatibleConjugacyClasses 71.7 Operators for Character Tables 71.8 Attributes and Properties for Groups and Character Tables 71.8-1 CharacterDegrees 71.8-2 Irr 71.8-3 LinearCharacters 71.8-4 OrdinaryCharacterTable 71.8-5 Group Operations Applicable to Character Tables 71.9 Attributes and Properties only for Character Tables 71.9-1 OrdersClassRepresentatives 71.9-2 SizesCentralizers 71.9-3 SizesConjugacyClasses 71.9-4 AutomorphismsOfTable 71.9-5 UnderlyingCharacteristic 71.9-6 Class Names and Character Names 71.9-7 Class Parameters and Character Parameters 71.9-8 Identifier 71.9-9 InfoText 71.9-10 InverseClasses 71.9-11 RealClasses 71.9-12 ClassOrbit 71.9-13 ClassRoots 71.10 Normal Subgroups Represented by Lists of Class Positions 71.10-1 ClassPositionsOfNormalSubgroups 71.10-2 ClassPositionsOfAgemo 71.10-3 ClassPositionsOfCentre 71.10-4 ClassPositionsOfDirectProductDecompositions 71.10-5 ClassPositionsOfDerivedSubgroup 71.10-6 ClassPositionsOfElementaryAbelianSeries 71.10-7 ClassPositionsOfFittingSubgroup 71.10-8 ClassPositionsOfLowerCentralSeries 71.10-9 ClassPositionsOfUpperCentralSeries 71.10-10 ClassPositionsOfSupersolvableResiduum 71.10-11 ClassPositionsOfPCore 71.10-12 ClassPositionsOfNormalClosure 71.11 Operations Concerning Blocks 71.11-1 PrimeBlocks 71.11-2 SameBlock 71.11-3 BlocksInfo 71.11-4 DecompositionMatrix 71.11-5 LaTeXStringDecompositionMatrix 71.12 Other Operations for Character Tables 71.12-1 Index 71.12-2 IsInternallyConsistent 71.12-3 IsPSolvableCharacterTable 71.12-4 IsClassFusionOfNormalSubgroup 71.12-5 Indicator 71.12-6 NrPolyhedralSubgroups 71.12-7 ClassMultiplicationCoefficient 71.12-8 ClassStructureCharTable 71.12-9 MatClassMultCoeffsCharTable 71.13 Printing Character Tables 71.13-1 ViewObj 71.13-2 PrintObj 71.13-3 Display 71.13-4 DisplayOptions 71.13-5 PrintCharacterTable 71.14 Computing the Irreducible Characters of a Group 71.14-1 IrrDixonSchneider 71.14-2 IrrConlon 71.14-3 IrrBaumClausen 71.14-4 IrreducibleRepresentations 71.14-5 IrreducibleRepresentationsDixon 71.15 Representations Given by Modules 71.15-1 IrreducibleModules 71.15-2 AbsolutelyIrreducibleModules 71.15-3 RegularModule 71.16 The Dixon-Schneider Algorithm 71.17 Advanced Methods for Dixon-Schneider Calculations 71.17-1 DixonRecord 71.17-2 DixonInit 71.17-3 DixontinI 71.17-4 DixonSplit 71.17-5 BestSplittingMatrix 71.17-6 DxIncludeIrreducibles 71.17-7 SplitCharacters 71.17-8 IsDxLargeGroup 71.18 Components of a Dixon Record 71.19 An Example of Advanced Dixon-Schneider Calculations 71.20 Constructing Character Tables from Others 71.20-1 CharacterTableDirectProduct 71.20-2 FactorsOfDirectProduct 71.20-3 CharacterTableFactorGroup 71.20-4 CharacterTableIsoclinic 71.20-5 CharacterTableWreathSymmetric 71.21 Sorted Character Tables 71.21-1 CharacterTableWithSortedCharacters 71.21-2 SortedCharacters 71.21-3 CharacterTableWithSortedClasses 71.21-4 SortedCharacterTable 71.21-5 ClassPermutation 71.22 Automorphisms and Equivalence of Character Tables 71.22-1 MatrixAutomorphisms 71.22-2 TableAutomorphisms 71.22-3 TransformingPermutations 71.22-4 TransformingPermutationsCharacterTables 71.22-5 FamiliesOfRows 71.23 Storing Normal Subgroup Information 71.23-1 NormalSubgroupClassesInfo 71.23-2 ClassPositionsOfNormalSubgroup 71.23-3 NormalSubgroupClasses 71.23-4 FactorGroupNormalSubgroupClasses 72 Class Functions 72.1 Why Class Functions? 72.1-1 IsClassFunction 72.2 Basic Operations for Class Functions 72.2-1 UnderlyingCharacterTable 72.2-2 ValuesOfClassFunction 72.3 Comparison of Class Functions 72.4 Arithmetic Operations for Class Functions 72.4-1 Characteristic 72.4-2 ComplexConjugate 72.4-3 Order 72.5 Printing Class Functions 72.5-1 ViewObj 72.5-2 PrintObj 72.5-3 Display 72.6 Creating Class Functions from Values Lists 72.6-1 ClassFunction 72.6-2 VirtualCharacter 72.6-3 Character 72.6-4 ClassFunctionSameType 72.7 Creating Class Functions using Groups 72.7-1 TrivialCharacter 72.7-2 NaturalCharacter 72.7-3 PermutationCharacter 72.8 Operations for Class Functions 72.8-1 IsCharacter 72.8-2 IsVirtualCharacter 72.8-3 IsIrreducibleCharacter 72.8-4 DegreeOfCharacter 72.8-5 ScalarProduct 72.8-6 MatScalarProducts 72.8-7 Norm 72.8-8 ConstituentsOfCharacter 72.8-9 KernelOfCharacter 72.8-10 ClassPositionsOfKernel 72.8-11 CentreOfCharacter 72.8-12 ClassPositionsOfCentre 72.8-13 InertiaSubgroup 72.8-14 CycleStructureClass 72.8-15 IsTransitive 72.8-16 Transitivity 72.8-17 CentralCharacter 72.8-18 DeterminantOfCharacter 72.8-19 EigenvaluesChar 72.8-20 Tensored 72.9 Restricted and Induced Class Functions 72.9-1 RestrictedClassFunction 72.9-2 RestrictedClassFunctions 72.9-3 InducedClassFunction 72.9-4 InducedClassFunctions 72.9-5 InducedClassFunctionsByFusionMap 72.9-6 InducedCyclic 72.10 Reducing Virtual Characters 72.10-1 ReducedClassFunctions 72.10-2 ReducedCharacters 72.10-3 IrreducibleDifferences 72.10-4 LLL 72.10-5 Extract 72.10-6 OrthogonalEmbeddingsSpecialDimension 72.10-7 Decreased 72.10-8 DnLattice 72.10-9 DnLatticeIterative 72.11 Symmetrizations of Class Functions 72.11-1 Symmetrizations 72.11-2 SymmetricParts 72.11-3 AntiSymmetricParts 72.11-4 OrthogonalComponents 72.11-5 SymplecticComponents 72.12 Molien Series 72.12-1 MolienSeries 72.12-2 MolienSeriesInfo 72.12-3 ValueMolienSeries 72.12-4 MolienSeriesWithGivenDenominator 72.13 Possible Permutation Characters 72.13-1 PermCharInfo 72.13-2 PermCharInfoRelative 72.14 Computing Possible Permutation Characters 72.14-1 PermChars 72.14-2 TestPerm1, ..., TestPerm5 72.14-3 PermBounds 72.14-4 PermComb 72.14-5 Inequalities 72.15 Operations for Brauer Characters 72.15-1 FrobeniusCharacterValue 72.15-2 BrauerCharacterValue 72.15-3 SizeOfFieldOfDefinition 72.15-4 RealizableBrauerCharacters 72.16 Domains Generated by Class Functions 73 Maps Concerning Character Tables 73.1 Power Maps 73.1-1 PowerMap 73.1-2 PossiblePowerMaps 73.1-3 ElementOrdersPowerMap 73.1-4 PowerMapByComposition 73.2 Orbits on Sets of Possible Power Maps 73.2-1 OrbitPowerMaps 73.2-2 RepresentativesPowerMaps 73.3 Class Fusions between Character Tables 73.3-1 FusionConjugacyClasses 73.3-2 ComputedClassFusions 73.3-3 GetFusionMap 73.3-4 StoreFusion 73.3-5 NamesOfFusionSources 73.3-6 PossibleClassFusions 73.3-7 ConsiderStructureConstants 73.4 Orbits on Sets of Possible Class Fusions 73.4-1 OrbitFusions 73.4-2 RepresentativesFusions 73.5 Parametrized Maps 73.5-1 CompositionMaps 73.5-2 InverseMap 73.5-3 ProjectionMap 73.5-4 Indirected 73.5-5 Parametrized 73.5-6 ContainedMaps 73.5-7 UpdateMap 73.5-8 MeetMaps 73.5-9 CommutativeDiagram 73.5-10 CheckFixedPoints 73.5-11 TransferDiagram 73.5-12 TestConsistencyMaps 73.5-13 Indeterminateness 73.5-14 PrintAmbiguity 73.5-15 ContainedSpecialVectors 73.5-16 CollapsedMat 73.5-17 ContainedDecomposables 73.6 Subroutines for the Construction of Power Maps 73.6-1 InitPowerMap 73.6-2 Congruences 73.6-3 ConsiderKernels 73.6-4 ConsiderSmallerPowerMaps 73.6-5 MinusCharacter 73.6-6 PowerMapsAllowedBySymmetrizations 73.7 Subroutines for the Construction of Class Fusions 73.7-1 InitFusion 73.7-2 CheckPermChar 73.7-3 ConsiderTableAutomorphisms 73.7-4 FusionsAllowedByRestrictions 74 Unknowns 74.1 More about Unknowns 74.1-1 Unknown 74.1-2 LargestUnknown 74.1-3 IsUnknown 74.1-4 Comparison of Unknowns 74.1-5 Arithmetical Operations for Unknowns 75 Monomiality Questions 75.1 InfoMonomial (Info Class) 75.1-1 InfoMonomial 75.2 Character Degrees and Derived Length 75.2-1 Alpha 75.2-2 Delta 75.2-3 IsBergerCondition 75.3 Primitivity of Characters 75.3-1 TestHomogeneous 75.3-2 IsPrimitiveCharacter 75.3-3 TestQuasiPrimitive 75.3-4 TestInducedFromNormalSubgroup 75.4 Testing Monomiality 75.4-1 TestMonomial 75.4-2 TestMonomialUseLattice 75.4-3 IsMonomialNumber 75.4-4 TestMonomialQuick 75.4-5 TestSubnormallyMonomial 75.4-6 TestRelativelySM 75.5 Minimal Nonmonomial Groups 75.5-1 IsMinimalNonmonomial 75.5-2 MinimalNonmonomialGroup 76 Using GAP Packages 76.1 Installing a GAP Package 76.2 Loading a GAP Package 76.2-1 LoadPackage 76.2-2 SetPackagePath 76.2-3 ExtendRootDirectories 76.2-4 DisplayPackageLoadingLog 76.3 Functions for GAP Packages 76.3-1 ReadPackage 76.3-2 TestPackageAvailability 76.3-3 TestPackage 76.3-4 InstalledPackageVersion 76.3-5 DirectoriesPackageLibrary 76.3-6 DirectoriesPackagePrograms 76.3-7 CompareVersionNumbers 76.3-8 IsPackageMarkedForLoading 76.3-9 DeclareAutoreadableVariables 76.3-10 Kernel modules in GAP packages 76.3-11 LoadDynamicModule 76.3-12 The PackageInfo.g File 76.3-13 ValidatePackageInfo 76.3-14 ShowPackageVariables 76.3-15 BibEntry 76.3-16 Cite 77 Replaced and Removed Command Names 77.1 Group Actions – Name Changes 77.2 Package Interface – Obsolete Functions and Name Changes 77.3 Normal Forms of Integer Matrices – Name Changes 77.4 Miscellaneous Name Changes or Removed Names 77.4-1 InfoObsolete 77.5 The former .gaprc file 77.6 Semigroup properties 77.6-1 IsSemilatticeAsSemigroup 78 Method Selection 78.1 Operations and Methods 78.2 Method Installation 78.2-1 InstallMethod 78.2-2 InstallOtherMethod 78.3 Applicable Methods and Method Selection 78.4 Partial Methods 78.4-1 TryNextMethod 78.5 Redispatching 78.5-1 RedispatchOnCondition 78.6 Immediate Methods 78.6-1 InstallImmediateMethod 78.7 Logical Implications 78.7-1 InstallTrueMethod 78.8 Operations and Mathematical Terms 79 Creating New Objects 79.1 Creating Categories 79.1-1 NewCategory 79.1-2 CategoryFamily 79.2 Creating Representations 79.2-1 NewRepresentation 79.3 Creating Attributes and Properties 79.3-1 NewAttribute 79.3-2 NewProperty 79.4 Creating Other Filters 79.4-1 NewFilter 79.4-2 SetFilterObj 79.4-3 ResetFilterObj 79.5 Creating Operations 79.5-1 NewOperation 79.6 Creating Constructors 79.6-1 NewConstructor 79.7 Creating Families 79.7-1 NewFamily 79.8 Creating Types 79.8-1 NewType 79.9 Creating Objects 79.9-1 Objectify 79.9-2 ObjectifyWithAttributes 79.10 Component Objects 79.10-1 NamesOfComponents 79.11 Positional Objects 79.12 Implementing New List Objects 79.13 Example – Constructing Enumerators 79.14 Example – Constructing Iterators 79.15 Arithmetic Issues in the Implementation of New Kinds of Lists 79.16 External Representation 79.16-1 ExtRepOfObj 79.17 Mutability and Copying 79.18 Global Variables in the Library 79.18-1 DeclareCategory 79.18-2 DeclareRepresentation 79.18-3 DeclareAttribute 79.18-4 DeclareProperty 79.18-5 DeclareFilter 79.18-6 DeclareOperation 79.18-7 DeclareGlobalFunction 79.18-8 DeclareGlobalVariable 79.18-9 InstallValue 79.18-10 DeclareSynonym 79.18-11 FlushCaches 79.19 Declaration and Implementation Part 80 Examples of Extending the System 80.1 Addition of a Method 80.2 Extending the Range of Definition of an Existing Operation 80.3 Enforcing Property Tests 80.4 Adding a new Operation 80.5 Adding a new Attribute 80.6 Adding a new Representation 80.7 Components versus Attributes 80.8 Adding new Concepts 80.8-1 Example: M-groups 80.8-2 Example: Groups with a word length 80.8-3 Example: Groups with a decomposition as semidirect product 80.9 Creating Own Arithmetic Objects 80.9-1 ArithmeticElementCreator 80.9-2 Example: ArithmeticElementCreator 81 An Example – Residue Class Rings 81.1 A First Attempt to Implement Elements of Residue Class Rings 81.2 Why Proceed in a Different Way? 81.3 A Second Attempt to Implement Elements of Residue Class Rings 81.4 Compatibility of Residue Class Rings with Prime Fields 81.5 Further Improvements in Implementing Residue Class Rings 82 An Example – Designing Arithmetic Operations 82.1 New Arithmetic Operations vs. New Objects 82.2 Designing new Multiplicative Objects 83 Library Files 83.1 File Types 83.2 Finding Implementations in the Library 83.3 Undocumented Variables 84 Interface to the GAP Help System 84.1 Installing and Removing a Help Book 84.1-1 HELP_ADD_BOOK 84.1-2 HELP_REMOVE_BOOK 84.2 The manual.six File 84.3 The Help Book Handler 84.4 Introducing new Viewer for the Online Help 84.4-1 HELP_VIEWER_INFO 85 Function-Operation-Attribute Triples 85.1 Key Dependent Operations 85.1-1 KeyDependentOperation 85.2 In Parent Attributes 85.2-1 InParentFOA 85.3 Operation Functions 85.3-1 OrbitsishOperation 85.3-2 OrbitishFO 85.3-3 Example: Orbit and OrbitOp 86 Weak Pointers 86.1 Weak Pointer Objects 86.1-1 WeakPointerObj 86.2 Low Level Access Functions for Weak Pointer Objects 86.2-1 SetElmWPObj 86.3 Accessing Weak Pointer Objects as Lists 86.4 Copying Weak Pointer Objects 86.5 The GASMAN Interface for Weak Pointer Objects 87 More about Stabilizer Chains 87.1 Generalized Conjugation Technique 87.2 The General Backtrack Algorithm with Ordered Partitions 87.2-1 Internal representation of ordered partitions 87.2-2 Functions for setting up an R-base 87.2-3 Refinement functions for the backtrack search 87.2-4 Functions for meeting ordered partitions 87.2-5 Avoiding multiplication of permutations 87.3 Stabilizer Chains for Automorphisms Acting on Enumerators 87.3-1 An operation domain for automorphisms 87.3-2 Enumerators for cosets of characteristic factors 87.3-3 Making automorphisms act on such enumerators