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GAP - Reference Manual

Release 4.13.1, 2024-06-11

The GAP Group
Email: support@gap-system.org
Homepage: https://www.gap-system.org

Copyright

Copyright © (1987-2024) 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 Version 2, see https://www.gnu.org/licenses/old-licenses/gpl-2.0.html or the LICENSE file in the root 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 has been 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

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
  3.2-5 User Preferences Defined by GAP
 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
  3.4-4 ARCH_IS_WSL
 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
  4.6-2 Conventions about Identifiers
 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 MakeConstantGlobal
  4.9-5 ValueGlobal
  4.9-6 IsBoundGlobal
  4.9-7 UnbindGlobal
  4.9-8 BindGlobal
  4.9-9 NamesGVars
  4.9-10 NamesSystemGVars
  4.9-11 NamesUserGVars
 4.10 Namespaces for GAP packages
 4.11 Function
 4.12 Function Calls

  4.12-1 Function Call With Arguments
  4.12-2 Function Call With Options
 4.13 Comparisons
 4.14 Arithmetic Operators
 4.15 Statements

  4.15-1 Assignments
  4.15-2 Procedure Calls
  4.15-3 If
  4.15-4 While
  4.15-5 Repeat
  4.15-6 For
  4.15-7 Break
  4.15-8 Continue
  4.15-9 Return (With or without Value)
 4.16 Syntax Trees

  4.16-1 SyntaxTree
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 LocationFunc
  5.1-7 PageSource
 5.2 Calling a function with a list argument that is interpreted as several arguments

  5.2-1 CallFuncList
 5.3 Wrapping a function, so the values produced are cached

  5.3-1 MemoizePosIntFunction
 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 FunctionsFamily
 5.6 Naming Conventions
 5.7 Code annotations (pragmas)
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 GapExitCode
  6.7-3 QuitGap
  6.7-4 ForceQuitGap
  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
  7.3-6 TraceInternalMethods
 7.4 Info Functions

  7.4-1 NewInfoClass
  7.4-2 DeclareInfoClass
  7.4-3 SetInfoLevel
  7.4-4 InfoLevel
  7.4-5 ShowUsedInfoClasses
  7.4-6 Info
  7.4-7 Customizing Info (7.4-6) statements
  7.4-8 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 NanosecondsSinceEpoch
  7.6-4 time
  7.6-5 Sleep
 7.7 Tracking Memory Usage

  7.7-1 TotalMemoryAllocated
  7.7-2 memory_allocated
 7.8 Profiling

  7.8-1 Function Profiling
  7.8-2 ProfileGlobalFunctions
  7.8-3 ProfileOperations
  7.8-4 ProfileOperationsAndMethods
  7.8-5 ProfileFunctions
  7.8-6 UnprofileFunctions
  7.8-7 ProfileMethods
  7.8-8 UnprofileMethods
  7.8-9 DisplayProfile
  7.8-10 ClearProfile
  7.8-11 An Example of Function Profiling
  7.8-12 Line By Line Profiling
  7.8-13 Line by Line profiling example
  7.8-14 ProfileLineByLine
  7.8-15 CoverageLineByLine
  7.8-16 UnprofileLineByLine
  7.8-17 UncoverageLineByLine
  7.8-18 IsLineByLineProfileActive
  7.8-19 DisplayCacheStats
  7.8-20 ClearCacheStats
 7.9 Information about the version used
 7.10 Test Files

  7.10-1 Starting and stopping test
  7.10-2 Test
  7.10-3 TestDirectory
 7.11 Debugging Recursion

  7.11-1 SetRecursionTrapInterval
 7.12 Global Memory Information

  7.12-1 Garbage Collection
  7.12-2 CollectGarbage
  7.12-3 GasmanStatistics
  7.12-4 GasmanMessageStatus
  7.12-5 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 ChangeDirectoryCurrent
  9.3-6 DirectoriesLibrary
  9.3-7 DirectoriesSystemPrograms
  9.3-8 DirectoryContents
  9.3-9 DirectoryDesktop
  9.3-10 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 UserHomeExpand
  9.7-10 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
 10.12 Opening files in the Operating System

  10.12-1 OpenExternal
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
  13.1-2 NewFamily
 13.2 Filters

  13.2-1 RankFilter
  13.2-2 NamesFilter
  13.2-3 FilterByName
  13.2-4 ShowImpliedFilters
  13.2-5 FiltersType
 13.3 Categories

  13.3-1 IsCategory
  13.3-2 CategoriesOfObject
  13.3-3 CategoryByName
  13.3-4 NewCategory
  13.3-5 DeclareCategory
  13.3-6 CategoryFamily
 13.4 Representation

  13.4-1 Basic Representations of Objects
  13.4-2 IsRepresentation
  13.4-3 RepresentationsOfObject
  13.4-4 NewRepresentation
  13.4-5 DeclareRepresentation
 13.5 Attributes

  13.5-1 IsAttribute
  13.5-2 KnownAttributesOfObject
  13.5-3 NewAttribute
  13.5-4 DeclareAttribute
  13.5-5 IsAttributeStoringRep
 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 IsProperty
  13.7-2 KnownPropertiesOfObject
  13.7-3 KnownTruePropertiesOfObject
  13.7-4 NewProperty
  13.7-5 DeclareProperty
 13.8 Other Filters

  13.8-1 NewFilter
  13.8-2 DeclareFilter
  13.8-3 SetFilterObj
  13.8-4 ResetFilterObj
 13.9 Types

  13.9-1 TypeObj
  13.9-2 DataType
  13.9-3 NewType
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 IsSquareInt
  14.2-12 ListOfDigits
  14.2-13 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 and Reset for Random Sources
  14.7-4 Kinds of Random Sources
  14.7-5 RandomSource
  14.7-6 Implementing new kinds of random sources
 14.8 Bitfields

  14.8-1 MakeBitfields
  14.8-2 BuildBitfields
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 DLog
  15.3-4 PrimitiveRootMod
  15.3-5 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 PValuation
  15.7-2 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 IteratorOfPartitionsSet
  16.2-21 NrPartitions
  16.2-22 OrderedPartitions
  16.2-23 NrOrderedPartitions
  16.2-24 PartitionsGreatestLE
  16.2-25 PartitionsGreatestEQ
  16.2-26 RestrictedPartitions
  16.2-27 NrRestrictedPartitions
  16.2-28 SignPartition
  16.2-29 AssociatedPartition
  16.2-30 PowerPartition
  16.2-31 PartitionTuples
  16.2-32 NrPartitionTuples
  16.2-33 BetaSet
 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, \(\ldots\), EH
  18.4-2 EI and ER
  18.4-3 EY, EX, \(\ldots\), ES
  18.4-4 EM, EL, \(\ldots\), 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 Float creators
  19.2-2 Rat
  19.2-3 Cyc
  19.2-4 SetFloats
  19.2-5 FLOAT
  19.2-6 EqFloat
  19.2-7 PrecisionFloat
  19.2-8 SignBit
  19.2-9 SinCos
  19.2-10 Atan2
  19.2-11 Log1p
  19.2-12 Erf
  19.2-13 Infinity testers
  19.2-14 Standard mathematical operations
 19.3 High-precision-specific methods
 19.4 Complex arithmetic

  19.4-1 Argument
 19.5 Interval-specific methods

  19.5-1 Sup
  19.5-2 Inf
  19.5-3 Mid
  19.5-4 AbsoluteDiameter
  19.5-5 RelativeDiameter
  19.5-6 IsDisjoint
  19.5-7 IsSubset
  19.5-8 IncreaseInterval
  19.5-9 BlowupInterval
  19.5-10 BisectInterval
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 GetWithDefault
  21.5-3 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 PositionSortedBy
  21.16-7 PositionSet
  21.16-8 PositionMaximum
  21.16-9 PositionProperty
  21.16-10 PositionsProperty
  21.16-11 PositionBound
  21.16-12 PositionsBound
  21.16-13 PositionNot
  21.16-14 PositionNonZero
  21.16-15 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 Apply
  21.20-10 Perform
  21.20-11 PermListList
  21.20-12 Maximum
  21.20-13 Minimum
  21.20-14 MaximumList and MinimumList
  21.20-15 Cartesian
  21.20-16 IteratorOfCartesianProduct
  21.20-17 Permuted
  21.20-18 List
  21.20-19 Filtered
  21.20-20 Number
  21.20-21 First
  21.20-22 Last
  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 IsRangeRep
  21.22-3 ConvertToRangeRep
 21.23 Enumerators

  21.23-1 IsQuickPositionList
 21.24 Plain Lists

  21.24-1 PlainListCopy
  21.24-2 IsPlistRep
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
  22.4-5 FlipBlist
  22.4-6 SetAllBlist
  22.4-7 ClearAllBlist
 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 ImmutableVector
  23.3-3 NumberFFVector
 23.4 Coefficient List Arithmetic

  23.4-1 AddVector
  23.4-2 AddCoeffs
  23.4-3 MultVector
  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 TraceMatrix
  24.4-4 DeterminantMatrix
  24.4-5 DeterminantMatrixDestructive
  24.4-6 DeterminantMatrixDivFree
  24.4-7 IsEmptyMatrix
  24.4-8 IsMonomialMatrix
  24.4-9 IsDiagonalMatrix
  24.4-10 IsUpperTriangularMatrix
  24.4-11 IsLowerTriangularMatrix
 24.5 Matrix Constructions

  24.5-1 IdentityMat
  24.5-2 NullMat
  24.5-3 EmptyMatrix
  24.5-4 DiagonalMat
  24.5-5 DiagonalMatrix
  24.5-6 PermutationMat
  24.5-7 TransposedMatImmutable
  24.5-8 TransposedMatDestructive
  24.5-9 KroneckerProduct
  24.5-10 ReflectionMat
  24.5-11 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 RankMatrix
  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 DiagonalOfMatrix
  24.12-2 UpperSubdiagonal
  24.12-3 DepthOfUpperTriangularMatrix
 24.13 Matrices as Linear Mappings

  24.13-1 CharacteristicPolynomial
  24.13-2 RationalCanonicalFormTransform
  24.13-3 JordanDecomposition
  24.13-4 BlownUpMat
  24.13-5 BlownUpVector
  24.13-6 CompanionMatrix
 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 BasisNullspaceModN
  24.15-3 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
 24.18 Linear Programming

  24.18-1 SimplexMethod
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 Concepts and Rules for Vector and Matrix Objects
 26.2 Categories of Vector and Matrix Objects

  26.2-1 IsVectorObj
  26.2-2 IsMatrixObj
  26.2-3 IsMatrixOrMatrixObj
  26.2-4 IsRowListMatrix
 26.3 Defining Attributes of Vector and Matrix Objects

  26.3-1 BaseDomain
  26.3-2 ConstructingFilter
  26.3-3 CompatibleVectorFilter
  26.3-4 Length
  26.3-5 NumberRows and NumberColumns
 26.4 Constructing Vector and Matrix Objects

  26.4-1 NewVector and NewZeroVector
  26.4-2 Vector
  26.4-3 ZeroVector
  26.4-4 NewMatrix, NewZeroMatrix, NewIdentityMatrix
  26.4-5 Matrix
  26.4-6 ZeroMatrix
  26.4-7 IdentityMatrix
 26.5 Operations for Base Domains of Vector and Matrix Objects

  26.5-1 OneOfBaseDomain and ZeroOfBaseDomain
 26.6 Operations for Vector and Matrix Objects

  26.6-1 Comparison of Vector and Matrix Objects
  26.6-2 Unpack
  26.6-3 ChangedBaseDomain
  26.6-4 Randomize
 26.7 List Like Operations for Vector Objects

  26.7-1 Element Access and Assignment for Vector Objects
  26.7-2 PositionNonZero
  26.7-3 PositionLastNonZero
  26.7-4 ListOp
 26.8 Arithmetical Operations for Vector Objects

  26.8-1 Unary Arithmetical Operations for Vector Objects
  26.8-2 Binary Arithmetical Operations for Vector Objects
  26.8-3 AddVector
  26.8-4 MultVector
 26.9 Operations for Vector Objects

  26.9-1 ConcatenationOfVectors
  26.9-2 ExtractSubVector
  26.9-3 CopySubVector
  26.9-4 WeightOfVector
  26.9-5 DistanceOfVectors
 26.10 Arithmetical Operations for Matrix Objects

  26.10-1 Unary Arithmetical Operations for Matrix Objects
  26.10-2 Binary Arithmetical Operations for Matrix Objects
 26.11 Operations for Matrix Objects

  26.11-1 MatElm
  26.11-2 SetMatElm
  26.11-3 ExtractSubMatrix
  26.11-4 MutableCopyMatrix
  26.11-5 CopySubMatrix
  26.11-6 CompatibleVector
  26.11-7 RowsOfMatrix
  26.11-8 CompanionMatrix
 26.12 Operations for Row List Matrix Objects

  26.12-1 List Access for a Row List Matrix
  26.12-2 List Assignment for a Row List Matrix
  26.12-3 Sublist Access for a Row List Matrix
  26.12-4 Sublist Assignment for a Row List Matrix
  26.12-5 IsBound\[\]
  26.12-6 Unbind\[\]
  26.12-7 Add
  26.12-8 Remove
  26.12-9 Append
  26.12-10 ShallowCopy
  26.12-11 ListOp
 26.13 Basic operations for row/column reductions

  26.13-1 MultMatrixRowLeft
  26.13-2 MultMatrixRowRight
  26.13-3 MultMatrixColumnRight
  26.13-4 MultMatrixColumnLeft
  26.13-5 AddMatrixRowsLeft
  26.13-6 AddMatrixRowsRight
  26.13-7 AddMatrixColumnsRight
  26.13-8 AddMatrixColumnsLeft
  26.13-9 SwapMatrixRows
  26.13-10 SwapMatrixColumns
 26.14 Implementing New Vector and Matrix Objects Types
 26.15 Available Representations of Vector Objects

  26.15-1 IsGF2VectorRep
  26.15-2 Is8BitVectorRep
  26.15-3 IsPlistVectorRep
  26.15-4 IsZmodnZVectorRep
 26.16 Available Representations of Matrix Objects

  26.16-1 IsGF2MatrixRep
  26.16-2 Is8BitMatrixRep
  26.16-3 IsPlistMatrixRep
  26.16-4 IsZmodnZMatrixRep
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 LowercaseChar
  27.7-13 UppercaseString
  27.7-14 UppercaseChar
  27.7-15 SplitString
  27.7-16 ReplacedString
  27.7-17 NormalizeWhitespace
  27.7-18 NormalizedWhitespace
  27.7-19 RemoveCharacters
  27.7-20 JoinStringsWithSeparator
  27.7-21 Chomp
  27.7-22 StartsWith
  27.7-23 StringFormatted
  27.7-24 NumbersString
  27.7-25 StringNumbers
  27.7-26 StringOfMemoryAmount
 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 Rat
  27.9-3 IntHexString
  27.9-4 Ordinal
  27.9-5 EvalString
  27.9-6 CrcString
  27.9-7 HexSHA256
  27.9-8 Pluralize
 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
  30.2-5 DeclareCategoryCollections
 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 Direct Products and their Elements

  32.1-1 IsDirectProductElement
  32.1-2 DirectProductFamily
 32.2 Creating Mappings

  32.2-1 GeneralMappingByElements
  32.2-2 MappingByFunction
  32.2-3 InverseGeneralMapping
  32.2-4 RestrictedInverseGeneralMapping
  32.2-5 CompositionMapping
  32.2-6 CompositionMapping2
  32.2-7 IsCompositionMappingRep
  32.2-8 ConstituentsCompositionMapping
  32.2-9 ZeroMapping
  32.2-10 IdentityMapping
  32.2-11 Embedding
  32.2-12 Projection
  32.2-13 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 IsBiCoset
  39.7-6 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 SolvableRadical
  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 IsQuasisimpleGroup
  39.15-13 IsomorphismTypeInfoFiniteSimpleGroup
  39.15-14 SimpleGroup
  39.15-15 SimpleGroupsIterator
  39.15-16 SmallSimpleGroup
  39.15-17 AllSmallNonabelianSimpleGroups
  39.15-18 IsFinitelyGeneratedGroup
  39.15-19 IsSubsetLocallyFiniteGroup
  39.15-20 IsPGroup
  39.15-21 IsPowerfulPGroup
  39.15-22 IsRegularPGroup
  39.15-23 PrimePGroup
  39.15-24 PClassPGroup
  39.15-25 RankPGroup
  39.15-26 IsPSolvable
  39.15-27 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
  39.17-19 StructuralSeriesOfGroup
 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 MaximalSubgroupClassReps
  39.19-6 LowIndexSubgroups
  39.19-7 AllSubgroups
  39.19-8 MaximalSubgroups
  39.19-9 NormalSubgroups
  39.19-10 MaximalNormalSubgroups
  39.19-11 MinimalNormalSubgroups
  39.19-12 CharacteristicSubgroups
 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 LowLayerSubgroups
  39.20-7 ContainedConjugates
  39.20-8 ContainingConjugates
  39.20-9 MinimalFaithfulPermutationDegree
  39.20-10 RepresentativesPerfectSubgroups
  39.20-11 ConjugacyClassesPerfectSubgroups
  39.20-12 Zuppos
  39.20-13 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 2-Cohomology

  39.25-1 TwoCohomologyGeneric
  39.25-2 FpGroupCocycle
 39.26 Tests for the Availability of Methods

  39.26-1 CanEasilyTestMembership
  39.26-2 CanEasilyComputeWithIndependentGensAbelianGroup
  39.26-3 CanComputeSize
  39.26-4 CanComputeSizeAnySubgroup
  39.26-5 CanComputeIndex
  39.26-6 CanComputeIsSubset
  39.26-7 KnowsHowToDecompose
 39.27 Specific functions for Normalizer calculation

  39.27-1 NormalizerViaRadical
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 InnerAutomorphismGroup
  40.7-7 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
  42.1-4 PERM_INVERSE_THRESHOLD
 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 CycleFromList
  42.5-6 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
  49.4-5 ListWreathProductElement
  49.4-6 WreathProductElementList
 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 IsDihedralGroup
  50.1-8 DicyclicGroup
  50.1-9 IsGeneralisedQuaternionGroup
  50.1-10 ExtraspecialGroup
  50.1-11 AlternatingGroup
  50.1-12 SymmetricGroup
  50.1-13 MathieuGroup
  50.1-14 SuzukiGroup
  50.1-15 ReeGroup
  50.1-16 Generator Names
 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 ProjectiveGeneralOrthogonalGroup
  50.2-17 ProjectiveSpecialOrthogonalGroup
  50.2-18 ProjectiveOmega
  50.2-19 ProjectiveGeneralSemilinearGroup
  50.2-20 ProjectiveSpecialSemilinearGroup
 50.3 Conjugacy Classes in Classical Groups

  50.3-1 NrConjugacyClassesGL
 50.4 Constructors for Basic Groups
 50.5 Selection Functions
 50.6 Finite Perfect Groups

  50.6-1 SizesPerfectGroups
  50.6-2 PerfectGroup
  50.6-3 PerfectIdentification
  50.6-4 NumberPerfectGroups
  50.6-5 SizeNumbersPerfectGroups
  50.6-6 DisplayInformationPerfectGroups
  50.6-7 More about the Perfect Groups Library
 50.7 Irreducible Maximal Finite Integral Matrix Groups

  50.7-1 ImfNumberQQClasses
  50.7-2 DisplayImfInvariants
  50.7-3 ImfInvariants
  50.7-4 ImfMatrixGroup
  50.7-5 IsomorphismPermGroup
  50.7-6 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 on 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
  51.8-13 DisplaySemigroup
 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 IsFpSemigroup
  52.1-3 IsElementOfFpSemigroup
  52.1-4 FpGrpMonSmgOfFpGrpMonSmgElement
 52.2 Creating Finitely Presented Semigroups and Monoids

  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 or Monoid

  52.4-1 UnderlyingElement
  52.4-2 ElementOfFpSemigroup
  52.4-3 FreeSemigroupOfFpSemigroup
  52.4-4 FreeGeneratorsOfFpSemigroup
  52.4-5 RelationsOfFpSemigroup
 52.5 Rewriting Systems and the Knuth-Bendix Procedure

  52.5-1 ReducedConfluentRewritingSystem
  52.5-2 KB_REW
  52.5-3 KnuthBendixRewritingSystem
  52.5-4 SemigroupOfRewritingSystem
  52.5-5 FreeSemigroupOfRewritingSystem
 52.6 Todd-Coxeter Procedure

  52.6-1 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 \^
  53.4-2 \^
  53.4-3 \*
  53.4-4 \/
  53.4-5 LeftQuotient
  53.4-6 \<
  53.4-7 \=
  53.4-8 PermLeftQuoTransformation
  53.4-9 IsInjectiveListTrans
  53.4-10 ComponentTransformationInt
  53.4-11 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 MultiplicativeZero
 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 Inverse
  54.5-2 \^
  54.5-3 \/
  54.5-4 \^
  54.5-5 \*
  54.5-6 \/
  54.5-7 LeftQuotient
  54.5-8 \<
  54.5-9 \=
  54.5-10 PermLeftQuoPartialPerm
  54.5-11 PreImagePartialPerm
  54.5-12 ComponentPartialPermInt
  54.5-13 NaturalLeqPartialPerm
  54.5-14 ShortLexLeqPartialPerm
  54.5-15 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 IsomorphismPartialPermSemigroup
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
  59.2-7 RootFFE
 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
 61.13 Tensor Products and Exterior and Symmetric Powers

  61.13-1 TensorProduct
  61.13-2 ExteriorPower
  61.13-3 SymmetricPower
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 AlgebraWithOneByStructureConstants
  62.4-3 StructureConstantsTable
  62.4-4 EmptySCTable
  62.4-5 SetEntrySCTable
  62.4-6 GapInputSCTable
  62.4-7 TestJacobi
  62.4-8 IdentityFromSCTable
  62.4-9 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 AlgebraHomomorphismByFunction
  62.10-8 NaturalHomomorphismByIdeal
  62.10-9 OperationAlgebraHomomorphism
  62.10-10 NiceAlgebraMonomorphism
  62.10-11 IsomorphismFpAlgebra
  62.10-12 IsomorphismMatrixAlgebra
  62.10-13 IsomorphismSCAlgebra
  62.10-14 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 of Algebra Modules

  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
  66.14-2 TaylorSeriesRationalFunction
 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
 67.3 Finding Subfields

  67.3-1 IdealDecompositionsOfPolynomial
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.InducedActionSubMatrix
  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 ClassPositionsOfSolvableRadical
  71.10-11 ClassPositionsOfSupersolvableResiduum
  71.10-12 ClassPositionsOfPCore
  71.10-13 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 CharacterTableOfNormalSubgroup
  71.20-6 CharacterTableWreathSymmetric
  71.20-7 CharacterValueWreathSymmetric
 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
  72.8-21 TensorProduct
 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 ExteriorPower
  72.11-5 SymmetricPower
  72.11-6 OrthogonalComponents
  72.11-7 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 and Developing GAP Packages
 76.1 Installing a GAP Package
 76.2 Loading a GAP Package

  76.2-1 LoadPackage
  76.2-2 Automatic loading of GAP packages
  76.2-3 SetPackagePath
  76.2-4 ExtendRootDirectories
  76.2-5 DisplayPackageLoadingLog
 76.3 Functions for GAP Packages

  76.3-1 ReadPackage
  76.3-2 TestPackageAvailability
  76.3-3 IsPackageLoaded
  76.3-4 IsPackageMarkedForLoading
  76.3-5 TestPackage
  76.3-6 InstalledPackageVersion
  76.3-7 DirectoriesPackageLibrary
  76.3-8 DirectoriesPackagePrograms
  76.3-9 CompareVersionNumbers
  76.3-10 DeclareAutoreadableVariables
  76.3-11 Kernel modules in GAP packages
  76.3-12 IsKernelExtensionAvailable
  76.3-13 LoadKernelExtension
  76.3-14 LoadDynamicModule
  76.3-15 The PackageInfo.g File
  76.3-16 ValidatePackageInfo
  76.3-17 ShowPackageVariables
  76.3-18 BibEntry
  76.3-19 Cite
 76.4 Guidelines for Writing a GAP Package
 76.5 Structure of a GAP Package
 76.6 Writing Documentation and Tools Needed
 76.7 An Example of a GAP Package
 76.8 File Structure
 76.9 Creating the PackageInfo.g File
 76.10 Functions and Variables and Choices of Their Names
 76.11 Package Dependencies (Requesting one GAP Package from within Another)
 76.12 Extensions Provided by a Package
 76.13 Declaration and Implementation Part of a Package
 76.14 Autoreadable Variables
 76.15 Standalone Programs in a GAP Package

  76.15-1 Installation of GAP Package Binaries
  76.15-2 Test for the Existence of GAP Package Binaries
  76.15-3 Calling of and Communication with External Binaries
 76.16 Having an InfoClass
 76.17 The Banner
 76.18 Version Numbers
 76.19 Testing a GAP package

  76.19-1 Tests files for a GAP package
  76.19-2 Testing GAP package loading
  76.19-3 LoadAllPackages
  76.19-4 Testing a GAP package with the GAP standard test suite
 76.20 Access to the GAP Development Version
 76.21 Version control and continuous integration for GAP packages
 76.22 Selecting a license for a GAP Package
 76.23 Releasing a GAP Package
 76.24 The homepage of a Package
 76.25 Some things to keep in mind
 76.26 Package release checklists

  76.26-1 Checklist for releasing a new package
  76.26-2 Checklist for upgrading the package for the next major release of GAP
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
78 Method Selection
 78.1 Operations and Methods

  78.1-1 IsOperation
  78.1-2 TypeOfOperation
  78.1-3 ShowDeclarationsOfOperation
  78.1-4 NewOperation
  78.1-5 DeclareOperation
  78.1-6 Tag Based Operations
 78.2 Constructors

  78.2-1 NewConstructor
  78.2-2 DeclareConstructor
 78.3 Method Installation

  78.3-1 InstallMethod
  78.3-2 InstallOtherMethod
  78.3-3 InstallEarlyMethod
  78.3-4 InstallMethodWithRandomSource
 78.4 Applicable Methods and Method Selection
 78.5 Partial Methods

  78.5-1 TryNextMethod
 78.6 Redispatching

  78.6-1 RedispatchOnCondition
 78.7 Immediate Methods

  78.7-1 InstallImmediateMethod
  78.7-2 IsNoImmediateMethodsObject
 78.8 Logical Implications

  78.8-1 InstallTrueMethod
  78.8-2 SuspendMethodReordering
 78.9 Operations and Mathematical Terms
79 Creating New Objects
 79.1 Creating Objects

  79.1-1 Objectify
  79.1-2 ObjectifyWithAttributes
 79.2 Component Objects

  79.2-1 NamesOfComponents
 79.3 Positional Objects
 79.4 Implementing New List Objects
 79.5 Example – Constructing Enumerators
 79.6 Example – Constructing Iterators
 79.7 Arithmetic Issues in the Implementation of New Kinds of Lists
 79.8 External Representation

  79.8-1 ExtRepOfObj
 79.9 Mutability and Copying
 79.10 Global Variables in the Library

  79.10-1 DeclareGlobalName
  79.10-2 DeclareGlobalVariable
  79.10-3 InstallValue
  79.10-4 FlushCaches
  79.10-5 DeclareGlobalFunction
  79.10-6 DeclareSynonym
 79.11 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
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
 87.3 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
References
Index

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