chap0.txt

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