C about.tex 1. About the GAP Reference Manual
S 1.1. Manual Conventions
S 1.2. Credit
C help.tex 2. The Help System
S 2.1. Invoking the Help
F 2.1. getting help
S 2.2. Browsing through the Sections
F 2.2. browsing forward
F 2.2. browsing backwards
F 2.2. browsing forward one chapter
F 2.2. browsing backwards one chapter
F 2.2. browsing the previous section browsed
F 2.2. browsing the next section browsed
F 2.2. list of available books
F 2.2. table of sections for help books
F 2.2. table of chapters for help books
F 2.2. redisplay a help section
F 2.2. redisplay with next help viewer
S 2.3. Changing the Help Viewer
I 2.3. document formats (text, dvi, ps, pdf, HTML)
F 2.3. SetHelpViewer
S 2.4. The Pager Command
F 2.4. Pager
C run.tex 3. Running GAP
I 3.0. options
S 3.1. Command Line Options
I 3.1. features!under UNIX
I 3.1. UNIX!features
I 3.1. options!under UNIX
I 3.1. UNIX!options
I 3.1. -h
I 3.1. -b
I 3.1. -q
I 3.1. -e
I 3.1. -f
I 3.1. -n
I 3.1. -x
I 3.1. -y
I 3.1. -g
I 3.1. -g -g
I 3.1. -m
I 3.1. -o
I 3.1. -K
I 3.1. -l
I 3.1. GAPInfo.RootPaths
I 3.1. -r
I 3.1. -L
I 3.1. -R
I 3.1. options!command line, filenames
S 3.2. Advanced Features of GAP
I 3.2. -a
I 3.2. -A
I 3.2. -B
I 3.2. -D
I 3.2. -M
I 3.2. -N
I 3.2. -O
I 3.2. -T
I 3.2. -X
I 3.2. -Y
I 3.2. -i
I 3.2. options!command line, internal
I 3.2. -C
I 3.2. -U
I 3.2. -P
I 3.2. -W
I 3.2. -z
S 3.3. Running GAP under MacOS
I 3.3. -z!on Macintosh
I 3.3. -P!on Macintosh
I 3.3. -W!on Macintosh
I 3.3. -a!on Macintosh
I 3.3. -f!on Macintosh
I 3.3. -n!on Macintosh
I 3.3. -e!on Macintosh
I 3.3. -o!on Macintosh
I 3.3. gap.rc
S 3.4. The .gaprc file
I 3.4. gap.rc
I 3.4. .gaprc
I 3.4. GAP3
S 3.5. Completion Files
F 3.5. CreateCompletionFiles
F 3.5. CreateCompletionFiles
S 3.6. Testing for the System Architecture
F 3.6. ARCH_IS_UNIX
F 3.6. ARCH_IS_MAC
F 3.6. ARCH_IS_WINDOWS
S 3.7. The Compiler
I 3.7. gac
F 3.7. LoadDynamicModule
F 3.7. LoadDynamicModule
S 3.8. Suitability for Compilation
S 3.9. Compiling Library Code
S 3.10. CRC Numbers
I 3.10. CRC
I 3.10. CrcFile!example
S 3.11. Saving and Loading a Workspace
I 3.11. save
F 3.11. SaveWorkspace
F 3.11. loading a saved workspace
C language.tex 4. The Programming Language
S 4.1. Language Overview
S 4.2. Lexical Structure
S 4.3. Symbols
S 4.4. Whitespaces
I 4.4. space
I 4.4. blank
I 4.4. tabulator
I 4.4. newline
I 4.4. comments
S 4.5. Keywords
S 4.6. Identifiers
F 4.6. IsValidIdentifier
S 4.7. Expressions
I 4.7. evaluation
I 4.7. operators
S 4.8. Variables
I 4.8. scope
I 4.8. bound
F 4.8. Unbind
S 4.9. More About Global Variables
F 4.9. IsReadOnlyGlobal
F 4.9. MakeReadOnlyGlobal
F 4.9. MakeReadWriteGlobal
F 4.9. ValueGlobal
F 4.9. IsBoundGlobal
F 4.9. UnbindGlobal
F 4.9. BindGlobal
F 4.9. NamesGVars
F 4.9. NamesSystemGVars
F 4.9. NamesUserGVars
F 4.9. TemporaryGlobalVarName
S 4.10. Function Calls
F 4.10. function call
F 4.10. function call!with arguments
I 4.10. functions!with a variable number of arguments
I 4.10. arg!special function argument
F 4.10. function call!with options
S 4.11. Comparisons
F 4.11. equality test
F 4.11. inequality test
F 4.11. smaller test
F 4.11. larger test
F 4.11. smaller or equal
F 4.11. larger or equal
I 4.11. operators!precedence
S 4.12. Arithmetic Operators
I 4.12. precedence
I 4.12. associativity
I 4.12. operators!arithmetic
I 4.12. +
I 4.12. -
I 4.12. \*
I 4.12. /
I 4.12. {\accent 94 }
I 4.12. mod!arithmetic operators
I 4.12. modulo
I 4.12. modulo!arithmetic operators
F 4.12. positive number
F 4.12. negative number
F 4.12. addition
F 4.12. subtraction
F 4.12. multiplication
F 4.12. division
F 4.12. mod
F 4.12. power
I 4.12. mod!rationals
I 4.12. modular remainder
I 4.12. modular inverse
I 4.12. coprime
I 4.12. relatively prime
I 4.12. operators!precedence
I 4.12. operators!associativity
S 4.13. Statements
I 4.13. execution
S 4.14. Assignments
F 4.14. assignment!variable
S 4.15. Procedure Calls
F 4.15. procedure call
F 4.15. procedure call with arguments
S 4.16. If
I 4.16. fi
I 4.16. then
I 4.16. else
I 4.16. elif
F 4.16. if statement
S 4.17. While
I 4.17. loop!while
F 4.17. while loop
S 4.18. Repeat
I 4.18. loop!repeat
I 4.18. until
F 4.18. repeat loop
S 4.19. For
I 4.19. loop!for
I 4.19. do
I 4.19. od
F 4.19. for loop
F 4.19. loop over range
F 4.19. loop over iterator
F 4.19. loop over object
S 4.20. Break
I 4.20. loops!leaving
F 4.20. break statement
S 4.21. Continue
I 4.21. loops!restarting
F 4.21. continue statement
S 4.22. Function
I 4.22. functions!definition of
I 4.22. end
I 4.22. local
I 4.22. recursion
I 4.22. functions!recursive
I 4.22. environment
I 4.22. body
F 4.22. function
I 4.22. functions!with a variable number of arguments
I 4.22. arg!special function argument
I 4.22. functions!definition by arrow notation
F 4.22. arrow notation for functions
S 4.23. Return
F 4.23. return!no value
F 4.23. return!with value
S 4.24. The Syntax in BNF
I 4.24. BNF
C function.tex 5. Functions
I 5.0. functions
S 5.1. Information about a function
F 5.1. NameFunction
F 5.1. NumberArgumentsFunction
S 5.2. Calling a function with a list argument that is interpreted as several arguments
F 5.2. CallFuncList
S 5.3. Functions that do nothing
F 5.3. ReturnTrue
F 5.3. ReturnFalse
F 5.3. ReturnFail
F 5.3. IdFunc
S 5.4. Function Types
F 5.4. IsFunction
F 5.4. IsOperation
F 5.4. FunctionsFamily
C mloop.tex 6. Main Loop and Break Loop
S 6.1. Main Loop
I 6.1. read eval print loop
I 6.1. loop!read eval print
I 6.1. prompt
I 6.1. prompt!partial
I 6.1. syntax errors
I 6.1. errors!syntax
I 6.1. output!suppressing
I 6.1. last
I 6.1. previous result
S 6.2. View and Print
F 6.2. View
F 6.2. Print
F 6.2. ViewObj
F 6.2. PrintObj
F 6.2. Display
S 6.3. Break Loops
F 6.3. quit
I 6.3. return
F 6.3. return from break loop
F 6.3. OnBreak
I 6.3. ErrorNoTraceBack
I 6.3. Break loop message
F 6.3. OnBreakMessage
I 6.3. Backtrace!GAP3 name for Where
F 6.3. Where
S 6.4. Variable Access in a Break Loop
F 6.4. DownEnv
F 6.4. UpEnv
S 6.5. Error
F 6.5. Error
S 6.6. ErrorCount
F 6.6. ErrorCount
S 6.7. Leaving GAP
I 6.7. quit!in emergency
I 6.7. exit
I 6.7. at exit functions
I 6.7. saving on exit
F 6.7. QUIT!emergency quit
F 6.7. InstallAtExit
F 6.7. QUITTING
F 6.7. SaveOnExitFile
S 6.8. Line Editing
S 6.9. Editing Files
F 6.9. Edit
S 6.10. Editor Support
I 6.10. utilities for editing GAP files
I 6.10. vi
I 6.10. vim
I 6.10. emacs
S 6.11. SizeScreen
F 6.11. SizeScreen
F 6.11. SizeScreen
C debug.tex 7. Debugging and Profiling Facilities
S 7.1. Recovery from NoMethodFound-Errors
F 7.1. ShowArguments
F 7.1. ShowArgument
F 7.1. ShowDetails
F 7.1. ShowMethods
F 7.1. ShowMethods
F 7.1. ShowOtherMethods
F 7.1. ShowOtherMethods
S 7.2. ApplicableMethod
F 7.2. ApplicableMethod
F 7.2. ApplicableMethod
F 7.2. ApplicableMethod
F 7.2. ApplicableMethodTypes
F 7.2. ApplicableMethodTypes
F 7.2. ApplicableMethodTypes
S 7.3. Tracing Methods
F 7.3. TraceMethods
F 7.3. UntraceMethods
F 7.3. TraceImmediateMethods
S 7.4. Info Functions
F 7.4. NewInfoClass
F 7.4. DeclareInfoClass
F 7.4. SetInfoLevel
F 7.4. InfoLevel
F 7.4. Info
F 7.4. InfoWarning
S 7.5. Assertions
F 7.5. SetAssertionLevel
F 7.5. AssertionLevel
F 7.5. Assert
F 7.5. Assert
S 7.6. Timing
F 7.6. Runtimes
F 7.6. Runtime
F 7.6. time
S 7.7. Profiling
F 7.7. ProfileOperations
F 7.7. ProfileOperationsAndMethods
F 7.7. ProfileMethods
F 7.7. UnprofileMethods
F 7.7. ProfileFunctions
F 7.7. UnprofileFunctions
F 7.7. ProfileGlobalFunctions
F 7.7. ProfileGlobalFunctions
F 7.7. DisplayProfile
F 7.7. DisplayProfile
F 7.7. PROFILETHRESHOLD
F 7.7. ClearProfile
F 7.7. DisplayCacheStats
F 7.7. ClearCacheStats
S 7.8. Information about the version used
F 7.8. DisplayRevision
S 7.9. Test Files
F 7.9. ReadTest
S 7.10. Debugging Recursion
F 7.10. SetRecursionTrapInterval
S 7.11. Global Memory Information
F 7.11. GasmanStatistics
F 7.11. GasmanMessageStatus
F 7.11. SetGasmanMessageStatus
F 7.11. GasmanLimits
C options.tex 8. Options Stack
F 8.0. PushOptions
F 8.0. PopOptions
F 8.0. ResetOptionsStack
F 8.0. ValueOption
F 8.0. DisplayOptionsStack
F 8.0. InfoOptions
C files.tex 9. Files and Filenames
S 9.1. Portability
F 9.1. LastSystemError
S 9.2. GAP Root Directory
S 9.3. Directories
F 9.3. Directory
F 9.3. DirectoryTemporary
F 9.3. DirectoryTemporary
F 9.3. DirectoryCurrent
F 9.3. DirectoriesLibrary
F 9.3. DirectoriesLibrary
F 9.3. DirectoriesSystemPrograms
F 9.3. DirectoryContents
S 9.4. Filename
F 9.4. Filename
F 9.4. Filename
S 9.5. Special Filenames
S 9.6. File Access
F 9.6. IsExistingFile
F 9.6. IsReadableFile
F 9.6. IsWritableFile
F 9.6. IsExecutableFile
F 9.6. IsDirectoryPath
S 9.7. File Operations
F 9.7. Read
F 9.7. ReadAsFunction
F 9.7. PrintTo
F 9.7. AppendTo
F 9.7. LogTo
F 9.7. LogTo!stop logging
F 9.7. InputLogTo
F 9.7. InputLogTo!stop logging input
F 9.7. OutputLogTo
F 9.7. OutputLogTo!stop logging output
F 9.7. CrcFile
F 9.7. RemoveFile
F 9.7. Reread
F 9.7. REREADING
C streams.tex 10. Streams
S 10.1. Categories for Streams and the StreamsFamily
F 10.1. IsStream
F 10.1. IsClosedStream
F 10.1. IsInputStream
F 10.1. IsInputTextStream
F 10.1. IsInputTextNone
F 10.1. IsOutputStream
F 10.1. IsOutputTextStream
F 10.1. IsOutputTextNone
F 10.1. StreamsFamily
S 10.2. Operations applicable to All Streams
F 10.2. CloseStream
F 10.2. FileDescriptorOfStream
F 10.2. UNIXSelect
S 10.3. Operations for Input Streams
F 10.3. Read!for streams
F 10.3. ReadAsFunction!for streams
F 10.3. ReadTest!for streams
F 10.3. ReadByte
F 10.3. ReadLine
F 10.3. ReadAll
F 10.3. ReadAll
F 10.3. IsEndOfStream
F 10.3. PositionStream
F 10.3. RewindStream
F 10.3. SeekPositionStream
S 10.4. Operations for Output Streams
F 10.4. WriteByte
F 10.4. WriteLine
F 10.4. WriteAll
F 10.4. PrintTo!for streams
F 10.4. AppendTo!for streams
F 10.4. LogTo!for streams
F 10.4. InputLogTo!for streams
F 10.4. OutputLogTo!for streams
F 10.4. SetPrintFormattingStatus
F 10.4. PrintFormattingStatus
S 10.5. File Streams
F 10.5. InputTextFile
F 10.5. OutputTextFile
S 10.6. User Streams
F 10.6. InputTextUser
F 10.6. OutputTextUser
S 10.7. String Streams
F 10.7. InputTextString
F 10.7. OutputTextString
S 10.8. Input-Output Streams
F 10.8. IsInputOutputStream
F 10.8. InputOutputLocalProcess
F 10.8. ReadAllLine
S 10.9. Dummy Streams
F 10.9. InputTextNone
F 10.9. OutputTextNone
S 10.10. Handling of Streams in the Background
F 10.10. InstallCharReadHookFunc
F 10.10. UnInstallCharReadHookFunc
C process.tex 11. Processes
S 11.1. Process
F 11.1. Process
S 11.2. Exec
F 11.2. Exec
C objects.tex 12. Objects and Elements
S 12.1. Objects
F 12.1. IsObject
S 12.2. Elements as equivalence classes
I 12.2. elements!definition
S 12.3. Sets
S 12.4. Domains
S 12.5. Identical Objects
F 12.5. IsIdenticalObj
F 12.5. IsNotIdenticalObj
S 12.6. Mutability and Copyability
F 12.6. IsCopyable
F 12.6. IsMutable
F 12.6. Immutable
F 12.6. MakeImmutable
S 12.7. Duplication of Objects
I 12.7. Copy
I 12.7. copy!an object
I 12.7. clone!an object
F 12.7. ShallowCopy
F 12.7. StructuralCopy
S 12.8. Other Operations Applicable to any Object
F 12.8. SetName
F 12.8. Name
F 12.8. IsInternallyConsistent
C types.tex 13. Types of Objects
S 13.1. Families
F 13.1. FamilyObj
S 13.2. Filters
I 13.2. and!for filters
F 13.2. RankFilter
F 13.2. NamesFilter
F 13.2. ShowImpliedFilters
S 13.3. Categories
F 13.3. CategoriesOfObject
S 13.4. Representation
F 13.4. RepresentationsOfObject
S 13.5. Attributes
I 13.5. system getter
I 13.5. system setter
F 13.5. KnownAttributesOfObject
S 13.6. Setter and Tester for Attributes
I 13.6. setter
I 13.6. tester
F 13.6. Tester
F 13.6. Setter
F 13.6. AttributeValueNotSet
F 13.6. InfoAttributes
F 13.6. DisableAttributeValueStoring
F 13.6. EnableAttributeValueStoring
S 13.7. Properties
F 13.7. KnownPropertiesOfObject
F 13.7. KnownTruePropertiesOfObject
S 13.8. Other Filters
S 13.9. Types
F 13.9. TypeObj
F 13.9. DataType
C integers.tex 14. Integers
F 14.0. Integers
F 14.0. PositiveIntegers
F 14.0. NonnegativeIntegers
F 14.0. IsIntegers
F 14.0. IsPositiveIntegers
F 14.0. IsNonnegativeIntegers
S 14.1. Elementary Operations for Integers
F 14.1. IsInt
F 14.1. IsPosInt
F 14.1. Int
F 14.1. IsEvenInt
F 14.1. IsOddInt
F 14.1. AbsInt
I 14.1. absolute value of an integer
F 14.1. SignInt
I 14.1. sign!of an integer
F 14.1. LogInt
F 14.1. RootInt
F 14.1. RootInt
I 14.1. root!of an integer
I 14.1. square root!of an integer
F 14.1. SmallestRootInt
I 14.1. root!of an integer, smallest
F 14.1. Random!for integers
S 14.2. Quotients and Remainders
F 14.2. QuoInt
I 14.2. integer part of a quotient
F 14.2. BestQuoInt
F 14.2. RemInt
I 14.2. remainder of a quotient
F 14.2. GcdInt
F 14.2. Gcdex
F 14.2. LcmInt
F 14.2. CoefficientsQadic
F 14.2. CoefficientsMultiadic
F 14.2. ChineseRem
I 14.2. Chinese remainder
F 14.2. PowerModInt
S 14.3. Prime Integers and Factorization
F 14.3. Primes
F 14.3. IsPrimeInt
F 14.3. IsProbablyPrimeInt
F 14.3. IsPrimePowerInt
F 14.3. NextPrimeInt
F 14.3. PrevPrimeInt
F 14.3. FactorsInt
F 14.3. FactorsInt
F 14.3. PrintFactorsInt
F 14.3. PrimePowersInt
F 14.3. DivisorsInt
I 14.3. divisors!of an integer
S 14.4. Residue Class Rings
I 14.4. mod!residue class rings
F 14.4. modulo!residue class rings
F 14.4. ZmodnZ
F 14.4. ZmodpZ
F 14.4. ZmodpZNC
I 14.4. mod!Integers
F 14.4. ZmodnZObj
F 14.4. ZmodnZObj
F 14.4. IsZmodnZObj
F 14.4. IsZmodnZObjNonprime
F 14.4. IsZmodpZObj
F 14.4. IsZmodpZObjSmall
F 14.4. IsZmodpZObjLarge
C numtheor.tex 15. Number Theory
I 15.0. prime residue group
F 15.0. InfoNumtheor
S 15.1. Prime Residues
F 15.1. PrimeResidues!function
I 15.1. prime residue group
F 15.1. Phi
I 15.1. order!of the prime residue group
I 15.1. prime residue group!order
I 15.1. Euler's totient function
F 15.1. Lambda
I 15.1. Carmichael's lambda function
I 15.1. prime residue group!exponent
I 15.1. exponent!of the prime residue group
F 15.1. GeneratorsPrimeResidues
S 15.2. Primitive Roots and Discrete Logarithms
F 15.2. OrderMod
I 15.2. multiplicative order of an integer
F 15.2. LogMod
F 15.2. LogModShanks
I 15.2. logarithm!discrete
F 15.2. PrimitiveRootMod
I 15.2. primitive root modulo an integer
I 15.2. prime residue group!generator
I 15.2. generator!of the prime residue group
F 15.2. IsPrimitiveRootMod
I 15.2. test!for a primitive root
I 15.2. prime residue group!generator
I 15.2. generator!of the prime residue group
S 15.3. Roots Modulo Integers
F 15.3. Jacobi
I 15.3. quadratic residue
I 15.3. residue!quadratic
F 15.3. Legendre
I 15.3. quadratic residue
I 15.3. residue!quadratic
F 15.3. RootMod
I 15.3. quadratic residue
I 15.3. residue!quadratic
I 15.3. root!of an integer modulo another
F 15.3. RootsMod
F 15.3. RootsUnityMod
I 15.3. modular roots
I 15.3. root!of 1 modulo an integer
S 15.4. Multiplicative Arithmetic Functions
F 15.4. Sigma
F 15.4. Tau
F 15.4. MoebiusMu
S 15.5. Miscellaneous
F 15.5. TwoSquares
I 15.5. representation!as a sum of two squares
C rational.tex 16. Rational Numbers
F 16.0. Rationals
F 16.0. IsRationals
S 16.1. Elementary Operations for Rationals
F 16.1. IsRat
I 16.1. test!for a rational
F 16.1. IsPosRat
F 16.1. IsNegRat
F 16.1. NumeratorRat
I 16.1. numerator!of a rational
F 16.1. DenominatorRat
I 16.1. denominator!of a rational
F 16.1. Rat
F 16.1. Random!for rationals
C combinat.tex 17. Combinatorics
S 17.1. Combinatorial Numbers
F 17.1. Factorial
F 17.1. Binomial
I 17.1. coefficient!binomial
I 17.1. number!binomial
F 17.1. Bell
I 17.1. number!Bell
F 17.1. Bernoulli
I 17.1. sequence!Bernoulli
F 17.1. Stirling1
I 17.1. Stirling number of the first kind
I 17.1. number!Stirling, of the first kind
F 17.1. Stirling2
I 17.1. Stirling number of the second kind
I 17.1. number!Stirling, of the second kind
S 17.2. Combinations, Arrangements and Tuples
F 17.2. Combinations
F 17.2. NrCombinations
I 17.2. powerset
I 17.2. subsets
F 17.2. Arrangements
F 17.2. NrArrangements
F 17.2. UnorderedTuples
F 17.2. NrUnorderedTuples
F 17.2. Tuples
F 17.2. NrTuples
F 17.2. PermutationsList
F 17.2. NrPermutationsList
F 17.2. Derangements
F 17.2. NrDerangements
F 17.2. PartitionsSet
F 17.2. NrPartitionsSet
F 17.2. Partitions
F 17.2. NrPartitions
F 17.2. OrderedPartitions
F 17.2. NrOrderedPartitions
I 17.2. partitions!ordered, of an integer
I 17.2. partitions!improper, of an integer
F 17.2. PartitionsGreatestLE
F 17.2. PartitionsGreatestEQ
F 17.2. RestrictedPartitions
F 17.2. NrRestrictedPartitions
I 17.2. partitions!restricted, of an integer
F 17.2. SignPartition
F 17.2. AssociatedPartition
F 17.2. PowerPartition
I 17.2. symmetric group!powermap
F 17.2. PartitionTuples
F 17.2. NrPartitionTuples
S 17.3. Fibonacci and Lucas Sequences
F 17.3. Fibonacci
I 17.3. sequence!Fibonacci
F 17.3. Lucas
I 17.3. sequence!Lucas
S 17.4. Permanent of a Matrix
F 17.4. Permanent
C cyclotom.tex 18. Cyclotomic Numbers
I 18.0. type!cyclotomic
I 18.0. irrationalities
I 18.0. cyclotomic field elements
S 18.1. Operations for Cyclotomics
I 18.1. roots of unity
F 18.1. E
F 18.1. Cyclotomics
F 18.1. IsCyclotomic
F 18.1. IsCyc
F 18.1. IsIntegralCyclotomic
I 18.1. Int!for cyclotomics
I 18.1. String!for cyclotomics
F 18.1. Conductor
F 18.1. Conductor
F 18.1. AbsoluteValue
F 18.1. RoundCyc
F 18.1. CoeffsCyc
I 18.1. coefficients!for cyclotomics
F 18.1. DenominatorCyc
F 18.1. ExtRepOfObj!external representation!for cyclotomics
F 18.1. DescriptionOfRootOfUnity
I 18.1. logarithm!of a root of unity
F 18.1. IsGaussInt
F 18.1. IsGaussRat
I 18.1. DefaultField!for cyclotomics
S 18.2. Infinity
F 18.2. IsInfinity
F 18.2. infinity
S 18.3. Comparisons of Cyclotomics
I 18.3. operators!for cyclotomics
S 18.4. ATLAS Irrationalities
I 18.4. atomic irrationalities
I 18.4. b_N
I 18.4. c_N
I 18.4. d_N
I 18.4. e_N
I 18.4. f_N
I 18.4. g_N
I 18.4. h_N
F 18.4. EB
F 18.4. EC
F 18.4. ED
F 18.4. EE
F 18.4. EF
F 18.4. EG
F 18.4. EH
I 18.4. i_N
I 18.4. r_N
F 18.4. EI
F 18.4. ER
I 18.4. s_N
I 18.4. t_N
I 18.4. u_N
I 18.4. v_N
I 18.4. w_N
I 18.4. x_N
I 18.4. y_N
F 18.4. EY
F 18.4. EX
F 18.4. EW
F 18.4. EV
F 18.4. EU
F 18.4. ET
F 18.4. ES
I 18.4. j_N
I 18.4. k_N
I 18.4. l_N
I 18.4. m_N
F 18.4. EM
F 18.4. EL
F 18.4. EK
F 18.4. EJ
I 18.4. n_k
F 18.4. NK
F 18.4. AtlasIrrationality
S 18.5. Galois Conjugacy of Cyclotomics
F 18.5. GaloisCyc
F 18.5. GaloisCyc
F 18.5. ComplexConjugate
F 18.5. StarCyc
F 18.5. Quadratic
F 18.5. GaloisMat
F 18.5. RationalizedMat
S 18.6. Internally Represented Cyclotomics
C unknown.tex 19. Unknowns
I 19.0. data type!unknown
F 19.0. Unknown
F 19.0. Unknown
F 19.0. LargestUnknown
F 19.0. IsUnknown
C boolean.tex 20. Booleans
I 20.0. type!boolean
I 20.0. logical
F 20.0. IsBool
S 20.1. Fail
F 20.1. fail
S 20.2. Comparisons of Booleans
F 20.2. comparisons!of booleans
F 20.2. comparisons!of booleans
F 20.2. ordering!booleans
S 20.3. Operations for Booleans
I 20.3. operations!for booleans
I 20.3. logical operations
F 20.3. or
F 20.3. and
F 20.3. and!for filters
F 20.3. not
C lists.tex 21. Lists
I 21.0. Sets
S 21.1. List Categories
F 21.1. IsList
F 21.1. IsDenseList
F 21.1. IsHomogeneousList
F 21.1. IsTable
F 21.1. IsConstantTimeAccessList
S 21.2. Basic Operations for Lists
F 21.2. list element!operation
F 21.2. list boundedness test!operation
F 21.2. list assignment!operation
F 21.2. list unbind!operation
S 21.3. List Elements
I 21.3. accessing!list elements
F 21.3. list element!access
F 21.3. sublist!access
I 21.3. sublist
F 21.3. sublist!operation
S 21.4. List Assignment
I 21.4. assignment!to a list
F 21.4. list element!assignment
F 21.4. sublist!assignment
F 21.4. sublist assignment!operation
F 21.4. Add
F 21.4. Append
S 21.5. IsBound and Unbind for Lists
F 21.5. IsBound!for lists
F 21.5. Unbind!for lists
S 21.6. Identical Lists
S 21.7. Duplication of Lists
I 21.7. ShallowCopy!for lists
I 21.7. StructuralCopy!for lists
S 21.8. Membership Test for Lists
I 21.8. in!for lists
F 21.8. element test!for lists
S 21.9. Enlarging Internally Represented Lists
S 21.10. Comparisons of Lists
I 21.10. comparisons!of lists
F 21.10. list equal!comparison
F 21.10. list smaller!comparison
S 21.11. Arithmetic for Lists
I 21.11. operators!for lists
S 21.12. Filters Controlling the Arithmetic Behaviour of Lists
F 21.12. IsGeneralizedRowVector
F 21.12. IsMultiplicativeGeneralizedRowVector
F 21.12. IsListDefault
F 21.12. NestingDepthA
F 21.12. NestingDepthM
S 21.13. Additive Arithmetic for Lists
I 21.13. addition!list and non-list
I 21.13. list and non-list!difference
S 21.14. Multiplicative Arithmetic for Lists
I 21.14. list and non-list!product
I 21.14. list and non-list!quotient
I 21.14. list and non-list!mod
I 21.14. mod!lists
I 21.14. list and non-list!left quotient
S 21.15. Mutability Status and List Arithmetic
F 21.15. ListWithIdenticalEntries
S 21.16. Finding Positions in Lists
F 21.16. Position
F 21.16. PositionCanonical
F 21.16. PositionNthOccurrence
F 21.16. PositionSorted
F 21.16. PositionSorted
F 21.16. PositionSet
F 21.16. PositionSet
F 21.16. PositionProperty
F 21.16. PositionBound
F 21.16. PositionNot
F 21.16. PositionNonZero
F 21.16. PositionSublist
F 21.16. PositionSublist
S 21.17. Properties and Attributes for Lists
F 21.17. IsMatchingSublist
F 21.17. IsMatchingSublist
F 21.17. IsDuplicateFree
F 21.17. IsDuplicateFreeList
I 21.17. duplicate free
F 21.17. IsSortedList
I 21.17. sorted list
F 21.17. IsSSortedList
F 21.17. IsSet
I 21.17. strictly sorted list
F 21.17. Length
F 21.17. ConstantTimeAccessList
S 21.18. Sorting Lists
F 21.18. Sort
F 21.18. Sort
F 21.18. SortParallel
F 21.18. SortParallel
F 21.18. Sortex
F 21.18. SortingPerm
S 21.19. Sorted Lists and Sets
I 21.19. sets
I 21.19. multisets
F 21.19. in!for strictly sorted lists
F 21.19. IsEqualSet
I 21.19. test!for set equality
F 21.19. IsSubsetSet
F 21.19. AddSet
I 21.19. add!an element to a set
F 21.19. RemoveSet
I 21.19. remove!an element from a set
F 21.19. UniteSet
I 21.19. union!of sets
F 21.19. IntersectSet
I 21.19. intersection!of sets
F 21.19. SubtractSet
I 21.19. subtract!a set from another
S 21.20. Operations for Lists
F 21.20. Concatenation
F 21.20. Concatenation
I 21.20. concatenation!of lists
F 21.20. Compacted
F 21.20. Collected
F 21.20. DuplicateFreeList
F 21.20. Unique
F 21.20. AsDuplicateFreeList
F 21.20. Flat
F 21.20. Reversed
F 21.20. IsLexicographicallyLess
F 21.20. Apply
F 21.20. Perform
F 21.20. PermListList
F 21.20. Maximum
F 21.20. Maximum
F 21.20. Minimum
F 21.20. Minimum
F 21.20. MaximumList
F 21.20. MinimumList
F 21.20. Cartesian
F 21.20. Cartesian
F 21.20. Permuted
F 21.20. List
F 21.20. List
F 21.20. List
F 21.20. Filtered
F 21.20. Filtered
F 21.20. Number
F 21.20. Number
F 21.20. Number
F 21.20. First
F 21.20. ForAll
F 21.20. ForAll
F 21.20. ForAny
F 21.20. ForAny
F 21.20. Product
F 21.20. Product
F 21.20. Product
F 21.20. Product
F 21.20. Sum
F 21.20. Sum
F 21.20. Sum
F 21.20. Sum
F 21.20. Iterated
F 21.20. ListN
S 21.21. Advanced List Manipulations
F 21.21. ListX
F 21.21. SetX
F 21.21. SumX
F 21.21. ProductX
S 21.22. Ranges
I 21.22. range
F 21.22. IsRange
F 21.22. ConvertToRangeRep
S 21.23. Enumerators
F 21.23. IsQuickPositionList
C blist.tex 22. Boolean Lists
F 22.0. IsBlist
S 22.1. Boolean Lists Representing Subsets
F 22.1. BlistList
F 22.1. ListBlist
F 22.1. SizeBlist
F 22.1. IsSubsetBlist
S 22.2. Set Operations via Boolean Lists
F 22.2. UnionBlist
F 22.2. UnionBlist
F 22.2. IntersectionBlist
F 22.2. IntersectionBlist
F 22.2. DifferenceBlist
S 22.3. Function that Modify Boolean Lists
F 22.3. UniteBlist
F 22.3. UniteBlistList
F 22.3. IntersectBlist
F 22.3. SubtractBlist
S 22.4. More about Boolean Lists
C vector.tex 23. Row Vectors
F 23.0. IsRowVector
S 23.1. Operators for Row Vectors
F 23.1. addition!vectors
F 23.1. addition!scalar and vector
F 23.1. addition!vector and scalar
F 23.1. subtraction!vectors
F 23.1. subtraction!scalar and vector
F 23.1. subtraction!vector and scalar
F 23.1. multiplication!scalar and vector
F 23.1. multiplication!vector and scalar
F 23.1. multiplication!vectors
F 23.1. NormedRowVector
S 23.2. Row Vectors over Finite Fields
F 23.2. ConvertToVectorRep
F 23.2. ConvertToVectorRep
F 23.2. ConvertToVectorRep
F 23.2. ConvertToVectorRepNC
F 23.2. ConvertToVectorRepNC
F 23.2. ConvertToVectorRepNC
F 23.2. NumberFFVector
S 23.3. Coefficient List Arithmetic
F 23.3. AddRowVector
F 23.3. AddCoeffs
F 23.3. AddCoeffs
F 23.3. AddCoeffs
F 23.3. MultRowVector
F 23.3. MultRowVector
F 23.3. CoeffsMod
S 23.4. Shifting and Trimming Coefficient Lists
F 23.4. LeftShiftRowVector
F 23.4. RightShiftRowVector
F 23.4. ShrinkRowVector
F 23.4. RemoveOuterCoeffs
S 23.5. Functions for Coding Theory
F 23.5. WeightVecFFE
F 23.5. DistanceVecFFE
F 23.5. DistancesDistributionVecFFEsVecFFE
F 23.5. DistancesDistributionMatFFEVecFFE
F 23.5. AClosestVectorCombinationsMatFFEVecFFE
F 23.5. CosetLeadersMatFFE
S 23.6. Vectors as coefficients of polynomials
F 23.6. ValuePol
F 23.6. ProductCoeffs
F 23.6. ReduceCoeffs
F 23.6. ReduceCoeffsMod
F 23.6. PowerModCoeffs
F 23.6. ShiftedCoeffs
F 23.6. ShrinkCoeffs
C matrix.tex 24. Matrices
F 24.0. InfoMatrix
S 24.1. Categories of Matrices
F 24.1. IsMatrix
F 24.1. IsOrdinaryMatrix
F 24.1. IsLieMatrix
S 24.2. Operators for Matrices
F 24.2. addition!matrices
F 24.2. addition!scalar and matrix
F 24.2. addition!matrix and scalar
F 24.2. subtraction!matrices
F 24.2. subtraction!scalar and matrix
F 24.2. subtraction!matrix and scalar
F 24.2. multiplication!scalar and matrix
F 24.2. multiplication!matrix and scalar
F 24.2. multiplication!vector and matrix
F 24.2. multiplication!matrix and vector
F 24.2. multiplication!matrices
F 24.2. inverse!matrix
F 24.2. quotient!matrices
F 24.2. quotient!scalar and matrix
F 24.2. quotient!matrix and scalar
F 24.2. quotient!vector and matrix
F 24.2. power!matrix
F 24.2. conjugate!matrix
F 24.2. image!vector under matrix
F 24.2. matrices!commutator
F 24.2. addition!scalar and matrix list
F 24.2. addition!scalar and matrix list
F 24.2. subtraction!scalar and matrix list
F 24.2. subtraction!scalar and matrix list
F 24.2. multiplication!scalar and matrix list
F 24.2. multiplication!scalar and matrix list
F 24.2. quotient!scalar and matrix list
F 24.2. multiplication!matrix and matrix list
F 24.2. multiplication!matrix and matrix list
F 24.2. quotient!matrix and matrix list
F 24.2. multiplication!vector and matrix list
S 24.3. Properties and Attributes of Matrices
F 24.3. DimensionsMat
F 24.3. DefaultFieldOfMatrix
I 24.3. Trace!of a matrix
F 24.3. TraceMat
F 24.3. Trace
F 24.3. DeterminantMat
F 24.3. Determinant
F 24.3. DeterminantMatDestructive
F 24.3. DeterminantMatDivFree
F 24.3. IsMonomialMatrix
F 24.3. IsDiagonalMat
F 24.3. IsUpperTriangularMat
F 24.3. IsLowerTriangularMat
S 24.4. Matrix Constructions
F 24.4. IdentityMat
F 24.4. NullMat
F 24.4. EmptyMatrix
F 24.4. DiagonalMat
F 24.4. PermutationMat
F 24.4. TransposedMatImmutable
F 24.4. TransposedMatAttr
F 24.4. TransposedMat
F 24.4. TransposedMatMutable
F 24.4. TransposedMatOp
F 24.4. TransposedMatDestructive
F 24.4. KroneckerProduct
F 24.4. ReflectionMat
F 24.4. ReflectionMat
F 24.4. ReflectionMat
F 24.4. ReflectionMat
F 24.4. PrintArray
F 24.4. MutableIdentityMat
F 24.4. MutableNullMat
S 24.5. Random Matrices
F 24.5. RandomMat
F 24.5. RandomInvertibleMat
F 24.5. RandomUnimodularMat
S 24.6. Matrices Representing Linear Equations and the Gaussian Algorithm
I 24.6. Gaussian algorithm
F 24.6. RankMat
F 24.6. TriangulizeMat
F 24.6. NullspaceMat
F 24.6. TriangulizedNullspaceMat
F 24.6. NullspaceMatDestructive
F 24.6. TriangulizedNullspaceMatDestructive
F 24.6. SolutionMat
F 24.6. SolutionMatDestructive
F 24.6. BaseFixedSpace
S 24.7. Eigenvectors and eigenvalues
F 24.7. GeneralisedEigenvalues
F 24.7. GeneralizedEigenvalues
F 24.7. GeneralisedEigenspaces
F 24.7. GeneralizedEigenspaces
F 24.7. Eigenvalues
F 24.7. Eigenspaces
F 24.7. Eigenvectors
S 24.8. Elementary Divisors
F 24.8. ElementaryDivisorsMat
F 24.8. ElementaryDivisorsMatDestructive
F 24.8. DiagonalizeMat
S 24.9. Echelonized Matrices
F 24.9. SemiEchelonMat
F 24.9. SemiEchelonMatDestructive
F 24.9. SemiEchelonMatTransformation
F 24.9. SemiEchelonMats
F 24.9. SemiEchelonMatsDestructive
S 24.10. Matrices as Basis of a Row Space
F 24.10. BaseMat
F 24.10. BaseMatDestructive
F 24.10. BaseOrthogonalSpaceMat
F 24.10. SumIntersectionMat
F 24.10. BaseSteinitzVectors
S 24.11. Triangular Matrices
F 24.11. DiagonalOfMat
F 24.11. UpperSubdiagonal
F 24.11. DepthOfUpperTriangularMatrix
S 24.12. Matrices as Linear Mappings
F 24.12. CharacteristicPolynomial
F 24.12. CharacteristicPolynomial
F 24.12. JordanDecomposition
F 24.12. BlownUpMat
F 24.12. BlownUpVector
F 24.12. CompanionMat
S 24.13. Matrices over Finite Fields
F 24.13. ImmutableMatrix
F 24.13. ConvertToMatrixRep
F 24.13. ConvertToMatrixRep
F 24.13. ConvertToMatrixRep
F 24.13. ConvertToMatrixRepNC
F 24.13. ConvertToMatrixRepNC
F 24.13. ConvertToMatrixRepNC
F 24.13. ProjectiveOrder
F 24.13. SimultaneousEigenvalues
F 24.13. InverseMatMod
F 24.13. NullspaceModQ
S 24.14. Special Multiplication Algorithms for Matrices over GF(2)
F 24.14. PROD_GF2MAT_GF2MAT_SIMPLE
F 24.14. PROD_GF2MAT_GF2MAT_ADVANCED
S 24.15. Block Matrices
I 24.15. IsBlockMatrixRep
F 24.15. AsBlockMatrix
F 24.15. BlockMatrix
F 24.15. BlockMatrix
F 24.15. MatrixByBlockMatrix
C matint.tex 25. Integral matrices and lattices
S 25.1. Linear equations over the integers and Integral Matrices
F 25.1. NullspaceIntMat
F 25.1. SolutionIntMat
F 25.1. SolutionNullspaceIntMat
F 25.1. BaseIntMat
F 25.1. BaseIntersectionIntMats
F 25.1. ComplementIntMat
S 25.2. Normal Forms over the Integers
F 25.2. TriangulizedIntegerMat
F 25.2. TriangulizedIntegerMatTransform
F 25.2. TriangulizeIntegerMat
F 25.2. HermiteNormalFormIntegerMat
F 25.2. HermiteNormalFormIntegerMatTransform
F 25.2. SmithNormalFormIntegerMat
F 25.2. SmithNormalFormIntegerMatTransforms
F 25.2. DiagonalizeIntMat
F 25.2. NormalFormIntMat
F 25.2. AbelianInvariantsOfList
S 25.3. Determinant of an integer matrix
F 25.3. DeterminantIntMat
S 25.4. Decompositions
I 25.4. decomposition matrix
I 25.4. DEC
F 25.4. Decomposition
F 25.4. Decomposition
F 25.4. LinearIndependentColumns
F 25.4. PadicCoefficients
F 25.4. IntegralizedMat
F 25.4. IntegralizedMat
F 25.4. DecompositionInt
S 25.5. Lattice Reduction
I 25.5. LLL algorithm!for vectors
I 25.5. short vectors spanning a lattice
I 25.5. lattice base reduction
F 25.5. LLLReducedBasis
I 25.5. LLL algorithm!for Gram matrices
I 25.5. lattice base reduction
F 25.5. LLLReducedGramMat
F 25.5. LLLReducedGramMat
S 25.6. Orthogonal Embeddings
F 25.6. OrthogonalEmbeddings
F 25.6. ShortestVectors
C string.tex 26. Strings and Characters
I 26.0. type!strings
I 26.0. doublequotes
I 26.0. singlequotes
F 26.0. IsChar
F 26.0. IsCharCollection
F 26.0. IsString
S 26.1. Special Characters
I 26.1. escaped characters
I 26.1. special character sequences
I 26.1. \\n
I 26.1. newline character
I 26.1. \\\"
I 26.1. doublequote character
I 26.1. \\'
I 26.1. singlequote character
I 26.1. \\\\
I 26.1. backslash character
I 26.1. \\b
I 26.1. backspace character
I 26.1. \\r
I 26.1. carriage return character
I 26.1. \\c
I 26.1. flush character
I 26.1. \\XYZ
I 26.1. octal character codes
I 26.1. escaping non-special characters
S 26.2. Internally Represented Strings
I 26.2. convert!to a string
F 26.2. IsStringRep
F 26.2. ConvertToStringRep
F 26.2. IsEmptyString
F 26.2. CharsFamily
S 26.3. Recognizing Characters
F 26.3. IsDigitChar
F 26.3. IsLowerAlphaChar
F 26.3. IsUpperAlphaChar
F 26.3. IsAlphaChar
S 26.4. Comparisons of Strings
F 26.4. strings!equality of
F 26.4. strings!inequality of
F 26.4. strings!lexicographic ordering of
S 26.5. Operations to Produce or Manipulate Strings
F 26.5. String
F 26.5. String
F 26.5. HexStringInt
F 26.5. StringPP
F 26.5. WordAlp
F 26.5. LowercaseString
F 26.5. SplitString
F 26.5. ReplacedString
F 26.5. NormalizeWhitespace
F 26.5. NormalizedWhitespace
F 26.5. JoinStringsWithSeparator
F 26.5. Chomp
S 26.6. Character Conversion
F 26.6. INT_CHAR
F 26.6. CHAR_INT
F 26.6. SINT_CHAR
F 26.6. CHAR_SINT
S 26.7. Operations to Evaluate Strings
I 26.7. evaluation!strings
F 26.7. Int!for strings
F 26.7. Rat!for strings
F 26.7. IntHexString
F 26.7. Ordinal
F 26.7. EvalString
S 26.8. Calendar Arithmetic
F 26.8. DaysInYear
F 26.8. DaysInMonth
F 26.8. DMYDay
F 26.8. DayDMY
F 26.8. WeekDay
F 26.8. StringDate
F 26.8. HMSMSec
F 26.8. SecHMSM
F 26.8. StringTime
F 26.8. SecondsDMYhms
F 26.8. DMYhmsSeconds
C record.tex 27. Records
I 27.0. type!records
F 27.0. IsRecord
F 27.0. IsRecordCollection
F 27.0. IsRecordCollColl
I 27.0. test!for records
F 27.0. RecNames
S 27.1. Accessing Record Elements
I 27.1. accessing!record elements
F 27.1. record!component access
F 27.1. record!component variable
S 27.2. Record Assignment
I 27.2. assignment!to a record
F 27.2. record!component assignment
F 27.2. record!component variable assignment
S 27.3. Identical Records
S 27.4. Comparisons of Records
F 27.4. equality!of records
F 27.4. inequality!of records
F 27.4. ordering!of records
F 27.4. ordering!of records
S 27.5. IsBound and Unbind for Records
S 27.6. Record Access Operations
F 27.6. NameRNam
F 27.6. RNamObj
F 27.6. RNamObj
F 27.6. record component!operation
F 27.6. record boundness test!operation
F 27.6. record assignment!operation
F 27.6. record unbind!operation
C coll.tex 28. Collections
F 28.0. IsCollection
S 28.1. Collection Families
F 28.1. CollectionsFamily
F 28.1. IsCollectionFamily
F 28.1. ElementsFamily
F 28.1. CategoryCollections
S 28.2. Lists and Collections
I 28.2. Sorted Lists as Collections
F 28.2. IsListOrCollection
F 28.2. Enumerator
F 28.2. Enumerator
F 28.2. EnumeratorSorted
F 28.2. EnumeratorSorted
F 28.2. EnumeratorByFunctions
F 28.2. EnumeratorByFunctions
F 28.2. SortedList
F 28.2. SortedList
F 28.2. SSortedList
F 28.2. SSortedList
F 28.2. Set
F 28.2. AsList
F 28.2. AsList
F 28.2. AsSortedList
F 28.2. AsSortedList
F 28.2. AsSSortedList
F 28.2. AsSSortedList
F 28.2. AsSet
I 28.2. elements!of a list or collection
F 28.2. Elements
S 28.3. Attributes and Properties for Collections
F 28.3. IsEmpty
F 28.3. IsEmpty
F 28.3. IsFinite
I 28.3. finiteness test!for a list or collection
F 28.3. IsTrivial
F 28.3. IsNonTrivial
F 28.3. IsWholeFamily
F 28.3. Size
F 28.3. Size
I 28.3. size!of a list or collection
I 28.3. order!of a list, collection or domain
F 28.3. Representative
I 28.3. representative!of a list or collection
F 28.3. RepresentativeSmallest
S 28.4. Operations for Collections
F 28.4. IsSubset
I 28.4. subset test!for collections
F 28.4. Intersection
F 28.4. Intersection
F 28.4. Intersection2
I 28.4. intersection!of collections
F 28.4. Union
F 28.4. Union
F 28.4. Union2
I 28.4. union!of collections
F 28.4. Difference
I 28.4. set difference!of collections
S 28.5. Membership Test for Collections
I 28.5. \\in!operation for testing membership
F 28.5. in!for collections
F 28.5. in!operation for
S 28.6. Random Elements
I 28.6. random element!of a list or collection
F 28.6. Random
F 28.6. Random
F 28.6. StateRandom
F 28.6. RestoreStateRandom
F 28.6. PseudoRandom
F 28.6. PseudoRandom
F 28.6. RandomList
I 28.6. random seed
S 28.7. Iterators
F 28.7. Iterator
F 28.7. Iterator
F 28.7. IteratorSorted
F 28.7. IteratorSorted
F 28.7. IsIterator
F 28.7. IsDoneIterator
F 28.7. NextIterator
F 28.7. IteratorList
F 28.7. TrivialIterator
F 28.7. IteratorByFunctions
C orders.tex 29. Orderings
F 29.0. IsOrdering
F 29.0. OrderingsFamily
S 29.1. Building new orderings
F 29.1. OrderingByLessThanFunctionNC
F 29.1. OrderingByLessThanFunctionNC
F 29.1. OrderingByLessThanOrEqualFunctionNC
F 29.1. OrderingByLessThanOrEqualFunctionNC
S 29.2. Properties and basic functionality
F 29.2. IsWellFoundedOrdering
F 29.2. IsTotalOrdering
F 29.2. IsIncomparableUnder
F 29.2. FamilyForOrdering
F 29.2. LessThanFunction
F 29.2. LessThanOrEqualFunction
F 29.2. IsLessThanUnder
F 29.2. IsLessThanOrEqualUnder
S 29.3. Orderings on families of associative words
F 29.3. IsOrderingOnFamilyOfAssocWords
F 29.3. IsTranslationInvariantOrdering
F 29.3. IsReductionOrdering
F 29.3. OrderingOnGenerators
F 29.3. LexicographicOrdering
F 29.3. LexicographicOrdering
F 29.3. LexicographicOrdering
F 29.3. LexicographicOrdering
F 29.3. LexicographicOrdering
F 29.3. LexicographicOrdering
F 29.3. ShortLexOrdering
F 29.3. ShortLexOrdering
F 29.3. ShortLexOrdering
F 29.3. ShortLexOrdering
F 29.3. ShortLexOrdering
F 29.3. ShortLexOrdering
F 29.3. IsShortLexOrdering
F 29.3. WeightLexOrdering
F 29.3. WeightLexOrdering
F 29.3. WeightLexOrdering
F 29.3. WeightLexOrdering
F 29.3. IsWeightLexOrdering
F 29.3. WeightOfGenerators
F 29.3. BasicWreathProductOrdering
F 29.3. BasicWreathProductOrdering
F 29.3. BasicWreathProductOrdering
F 29.3. BasicWreathProductOrdering
F 29.3. BasicWreathProductOrdering
F 29.3. BasicWreathProductOrdering
F 29.3. IsBasicWreathProductOrdering
F 29.3. WreathProductOrdering
F 29.3. WreathProductOrdering
F 29.3. WreathProductOrdering
F 29.3. WreathProductOrdering
F 29.3. WreathProductOrdering
F 29.3. WreathProductOrdering
F 29.3. IsWreathProductOrdering
F 29.3. LevelsOfGenerators
C domain.tex 30. Domains and their Elements
S 30.1. Operational Structure of Domains
S 30.2. Equality and Comparison of Domains
S 30.3. Constructing Domains
F 30.3. Struct
F 30.3. IsGeneratorsOfStruct
F 30.3. GeneratorsOfStruct
F 30.3. StructByGenerators
F 30.3. StructWithGenerators
F 30.3. ClosureStruct
S 30.4. Changing the Structure
F 30.4. AsStruct
S 30.5. Changing the Representation
F 30.5. IsomorphismRepStruct
S 30.6. Domain Categories
F 30.6. IsStruct
S 30.7. Parents
F 30.7. Parent
F 30.7. SetParent
F 30.7. HasParent
S 30.8. Constructing Subdomains
I 30.8. Subdomains
F 30.8. Substruct
F 30.8. SubstructNC
F 30.8. AsSubstruct
F 30.8. IsSubstruct
S 30.9. Operations for Domains
F 30.9. IsGeneralizedDomain
F 30.9. IsDomain
F 30.9. GeneratorsOfDomain
F 30.9. Domain
F 30.9. DomainByGenerators
S 30.10. Attributes and Properties of Elements
F 30.10. Characteristic
F 30.10. OneImmutable
F 30.10. OneAttr
F 30.10. One
F 30.10. Identity
F 30.10. OneMutable
F 30.10. OneOp
F 30.10. OneSameMutability
F 30.10. OneSM
F 30.10. ZeroImmutable
F 30.10. ZeroAttr
F 30.10. Zero
F 30.10. ZeroMutable
F 30.10. ZeroOp
F 30.10. ZeroSameMutability
F 30.10. ZeroSM
F 30.10. MultiplicativeZeroOp
F 30.10. IsOne
F 30.10. IsZero
F 30.10. IsIdempotent
F 30.10. InverseImmutable
F 30.10. InverseAttr
F 30.10. Inverse
F 30.10. InverseMutable
F 30.10. InverseOp
F 30.10. InverseSameMutability
F 30.10. InverseSM
F 30.10. AdditiveInverseImmutable
F 30.10. AdditiveInverseAttr
F 30.10. AdditiveInverse
F 30.10. AdditiveInverseMutable
F 30.10. AdditiveInverseOp
F 30.10. AdditiveInverseSameMutability
F 30.10. AdditiveInverseSM
F 30.10. Order
S 30.11. Comparison Operations for Elements
F 30.11. equality!operation
F 30.11. comparison!operation
F 30.11. CanEasilyCompareElements
F 30.11. CanEasilyCompareElementsFamily
F 30.11. CanEasilySortElements
F 30.11. CanEasilySortElementsFamily
S 30.12. Arithmetic Operations for Elements
F 30.12. addition!operation
F 30.12. multiplication!operation
F 30.12. division!operation
F 30.12. exponentiation!operation
F 30.12. remainder!operation
F 30.12. LeftQuotient
F 30.12. Comm
F 30.12. LieBracket
F 30.12. Sqrt
S 30.13. Relations Between Domains
F 30.13. UseSubsetRelation
F 30.13. UseIsomorphismRelation
F 30.13. UseFactorRelation
F 30.13. InstallSubsetMaintenance
F 30.13. InstallIsomorphismMaintenance
F 30.13. InstallFactorMaintenance
S 30.14. Useful Categories of Elements
F 30.14. IsExtAElement
F 30.14. IsNearAdditiveElement
F 30.14. IsAdditiveElement
F 30.14. IsNearAdditiveElementWithZero
F 30.14. IsAdditiveElementWithZero
F 30.14. IsNearAdditiveElementWithInverse
F 30.14. IsAdditiveElementWithInverse
F 30.14. IsExtLElement
F 30.14. IsExtRElement
F 30.14. IsMultiplicativeElement
F 30.14. IsMultiplicativeElementWithOne
F 30.14. IsMultiplicativeElementWithZero
F 30.14. IsMultiplicativeElementWithInverse
F 30.14. IsVector
F 30.14. IsNearRingElement
F 30.14. IsRingElement
F 30.14. IsNearRingElementWithOne
F 30.14. IsRingElementWithOne
F 30.14. IsNearRingElementWithInverse
F 30.14. IsRingElementWithInverse
F 30.14. IsScalar
S 30.15. Useful Categories for all Elements of a Family
F 30.15. IsAssociativeElement
F 30.15. IsAssociativeElementCollection
F 30.15. IsAssociativeElementCollColl
F 30.15. IsAdditivelyCommutativeElement
F 30.15. IsAdditivelyCommutativeElementCollection
F 30.15. IsAdditivelyCommutativeElementCollColl
F 30.15. IsAdditivelyCommutativeElementFamily
F 30.15. IsCommutativeElement
F 30.15. IsCommutativeElementCollection
F 30.15. IsCommutativeElementCollColl
F 30.15. IsFiniteOrderElement
F 30.15. IsFiniteOrderElementCollection
F 30.15. IsFiniteOrderElementCollColl
F 30.15. IsJacobianElement
F 30.15. IsJacobianElementCollection
F 30.15. IsJacobianElementCollColl
F 30.15. IsZeroSquaredElement
F 30.15. IsZeroSquaredElementCollection
F 30.15. IsZeroSquaredElementCollColl
C mapping.tex 31. Mappings
I 31.0. functions
I 31.0. relations
F 31.0. IsTuple
S 31.1. Creating Mappings
F 31.1. GeneralMappingByElements
F 31.1. MappingByFunction
F 31.1. MappingByFunction
F 31.1. MappingByFunction
F 31.1. InverseGeneralMapping
F 31.1. CompositionMapping
F 31.1. CompositionMapping2
F 31.1. IsCompositionMappingRep
F 31.1. ConstituentsCompositionMapping
F 31.1. ZeroMapping
F 31.1. IdentityMapping
F 31.1. Embedding
F 31.1. Embedding
F 31.1. Projection
F 31.1. Projection
F 31.1. Projection
F 31.1. RestrictedMapping
S 31.2. Properties and Attributes of (General) Mappings
F 31.2. IsTotal
F 31.2. IsSingleValued
F 31.2. IsMapping
F 31.2. IsInjective
F 31.2. IsSurjective
F 31.2. IsBijective
F 31.2. Range
F 31.2. Source
F 31.2. UnderlyingRelation
F 31.2. UnderlyingGeneralMapping
S 31.3. Images under Mappings
F 31.3. ImagesSource
F 31.3. ImagesRepresentative
F 31.3. ImagesElm
F 31.3. ImagesSet
F 31.3. ImageElm
F 31.3. Image
F 31.3. Image
F 31.3. Image
F 31.3. Images
F 31.3. Images
F 31.3. Images
S 31.4. Preimages under Mappings
F 31.4. PreImagesRange
F 31.4. PreImagesElm
F 31.4. PreImageElm
F 31.4. PreImagesRepresentative
F 31.4. PreImagesSet
F 31.4. PreImage
F 31.4. PreImage
F 31.4. PreImage
F 31.4. PreImages
F 31.4. PreImages
F 31.4. PreImages
S 31.5. Arithmetic Operations for General Mappings
S 31.6. Mappings which are Compatible with Algebraic Structures
S 31.7. Magma Homomorphisms
F 31.7. IsMagmaHomomorphism
F 31.7. MagmaHomomorphismByFunctionNC
F 31.7. NaturalHomomorphismByGenerators
S 31.8. Mappings that Respect Multiplication
F 31.8. RespectsMultiplication
F 31.8. RespectsOne
F 31.8. RespectsInverses
F 31.8. IsGroupGeneralMapping
F 31.8. IsGroupHomomorphism
F 31.8. KernelOfMultiplicativeGeneralMapping
F 31.8. CoKernelOfMultiplicativeGeneralMapping
S 31.9. Mappings that Respect Addition
F 31.9. RespectsAddition
F 31.9. RespectsAdditiveInverses
F 31.9. RespectsZero
F 31.9. IsAdditiveGroupGeneralMapping
F 31.9. IsAdditiveGroupHomomorphism
F 31.9. KernelOfAdditiveGeneralMapping
F 31.9. CoKernelOfAdditiveGeneralMapping
S 31.10. Linear Mappings
F 31.10. RespectsScalarMultiplication
F 31.10. IsLeftModuleGeneralMapping
F 31.10. IsLeftModuleHomomorphism
F 31.10. IsLinearMapping
S 31.11. Ring Homomorphisms
F 31.11. IsRingGeneralMapping
F 31.11. IsRingHomomorphism
F 31.11. IsRingWithOneGeneralMapping
F 31.11. IsRingWithOneHomomorphism
F 31.11. IsAlgebraGeneralMapping
F 31.11. IsAlgebraHomomorphism
F 31.11. IsAlgebraWithOneGeneralMapping
F 31.11. IsAlgebraWithOneHomomorphism
F 31.11. IsFieldHomomorphism
S 31.12. General Mappings
F 31.12. IsGeneralMapping
F 31.12. IsConstantTimeAccessGeneralMapping
F 31.12. IsEndoGeneralMapping
S 31.13. Technical Matters Concerning General Mappings
F 31.13. IsSPGeneralMapping
F 31.13. IsNonSPGeneralMapping
F 31.13. IsGeneralMappingFamily
F 31.13. FamilyRange
F 31.13. FamilySource
F 31.13. FamiliesOfGeneralMappingsAndRanges
F 31.13. GeneralMappingsFamily
F 31.13. TypeOfDefaultGeneralMapping
C relation.tex 32. Relations
I 32.0. binary relation
I 32.0. IsBinaryRelation!same as IsEndoGeneralMapping
I 32.0. IsEndoGeneralMapping!same as IsBinaryRelation
S 32.1. General Binary Relations
F 32.1. IsBinaryRelation
F 32.1. BinaryRelationByElements
F 32.1. IdentityBinaryRelation
F 32.1. IdentityBinaryRelation
F 32.1. EmptyBinaryRelation
F 32.1. EmptyBinaryRelation
S 32.2. Properties and Attributes of Binary Relations
F 32.2. IsReflexiveBinaryRelation
I 32.2. reflexive relation
F 32.2. IsSymmetricBinaryRelation
I 32.2. symmetric relation
F 32.2. IsTransitiveBinaryRelation
I 32.2. transitive relation
F 32.2. IsAntisymmetricBinaryRelation
I 32.2. antisymmetric relation
F 32.2. IsPreOrderBinaryRelation
I 32.2. preorder
F 32.2. IsPartialOrderBinaryRelation
I 32.2. partial order
F 32.2. IsHasseDiagram
F 32.2. IsEquivalenceRelation
I 32.2. equivalence relation
F 32.2. Successors
F 32.2. DegreeOfBinaryRelation
F 32.2. PartialOrderOfHasseDiagram
S 32.3. Binary Relations on Points
F 32.3. BinaryRelationOnPoints
F 32.3. BinaryRelationOnPointsNC
F 32.3. RandomBinaryRelationOnPoints
F 32.3. AsBinaryRelationOnPoints
F 32.3. AsBinaryRelationOnPoints
F 32.3. AsBinaryRelationOnPoints
S 32.4. Closure Operations and Other Constructors
F 32.4. ReflexiveClosureBinaryRelation
F 32.4. SymmetricClosureBinaryRelation
F 32.4. TransitiveClosureBinaryRelation
F 32.4. HasseDiagramBinaryRelation
F 32.4. StronglyConnectedComponents
F 32.4. PartialOrderByOrderingFunction
S 32.5. Equivalence Relations
I 32.5. equivalence relation
F 32.5. EquivalenceRelationByPartition
F 32.5. EquivalenceRelationByPartitionNC
F 32.5. EquivalenceRelationByRelation
F 32.5. EquivalenceRelationByPairs
F 32.5. EquivalenceRelationByPairsNC
F 32.5. EquivalenceRelationByProperty
S 32.6. Attributes of and Operations on Equivalence Relations
F 32.6. EquivalenceRelationPartition
F 32.6. GeneratorsOfEquivalenceRelationPartition
F 32.6. JoinEquivalenceRelations
F 32.6. MeetEquivalenceRelations
S 32.7. Equivalence Classes
F 32.7. IsEquivalenceClass
I 32.7. equivalence class
F 32.7. EquivalenceClassRelation
F 32.7. EquivalenceClasses!attribute
F 32.7. EquivalenceClassOfElement
F 32.7. EquivalenceClassOfElementNC
C magma.tex 33. Magmas
S 33.1. Magma Categories
F 33.1. IsMagma
F 33.1. IsMagmaWithOne
F 33.1. IsMagmaWithInversesIfNonzero
F 33.1. IsMagmaWithInverses
S 33.2. Magma Generation
F 33.2. Magma
F 33.2. Magma
F 33.2. MagmaWithOne
F 33.2. MagmaWithOne
F 33.2. MagmaWithInverses
F 33.2. MagmaWithInverses
F 33.2. MagmaByGenerators
F 33.2. MagmaByGenerators
F 33.2. MagmaWithOneByGenerators
F 33.2. MagmaWithOneByGenerators
F 33.2. MagmaWithInversesByGenerators
F 33.2. MagmaWithInversesByGenerators
F 33.2. Submagma
F 33.2. SubmagmaNC
F 33.2. SubmagmaWithOne
F 33.2. SubmagmaWithOneNC
F 33.2. SubmagmaWithInverses
F 33.2. SubmagmaWithInversesNC
F 33.2. AsMagma
F 33.2. AsSubmagma
F 33.2. InjectionZeroMagma
S 33.3. Magmas Defined by Multiplication Tables
F 33.3. MagmaByMultiplicationTable
F 33.3. MagmaWithOneByMultiplicationTable
F 33.3. MagmaWithInversesByMultiplicationTable
F 33.3. MagmaElement
F 33.3. MultiplicationTable
S 33.4. Attributes and Properties for Magmas
F 33.4. GeneratorsOfMagma
F 33.4. GeneratorsOfMagmaWithOne
F 33.4. GeneratorsOfMagmaWithInverses
I 33.4. centraliser
I 33.4. center
F 33.4. Centralizer
F 33.4. Centralizer
F 33.4. Centralizer
F 33.4. Centre
F 33.4. Center
F 33.4. Idempotents
F 33.4. IsAssociative
F 33.4. IsCentral
F 33.4. IsCommutative
F 33.4. IsAbelian
F 33.4. MultiplicativeNeutralElement
F 33.4. MultiplicativeZero
F 33.4. IsMultiplicativeZero
F 33.4. SquareRoots
F 33.4. TrivialSubmagmaWithOne
C word.tex 34. Words
S 34.1. Categories of Words and Nonassociative Words
I 34.1. abstract word
F 34.1. IsWord
F 34.1. IsWordWithOne
F 34.1. IsWordWithInverse
F 34.1. IsWordCollection
F 34.1. IsNonassocWord
F 34.1. IsNonassocWordWithOne
F 34.1. IsNonassocWordCollection
F 34.1. IsNonassocWordWithOneCollection
S 34.2. Comparison of Words
F 34.2. equality!nonassociative words
F 34.2. smaller!nonassociative words
S 34.3. Operations for Words
F 34.3. MappedWord
S 34.4. Free Magmas
F 34.4. FreeMagma
F 34.4. FreeMagma
F 34.4. FreeMagma
F 34.4. FreeMagma
F 34.4. FreeMagma
F 34.4. FreeMagmaWithOne
F 34.4. FreeMagmaWithOne
F 34.4. FreeMagmaWithOne
F 34.4. FreeMagmaWithOne
F 34.4. FreeMagmaWithOne
S 34.5. External Representation for Nonassociative Words
C wordass.tex 35. Associative Words
S 35.1. Categories of Associative Words
F 35.1. IsAssocWord
F 35.1. IsAssocWordWithOne
F 35.1. IsAssocWordWithInverse
S 35.2. Free Groups, Monoids and Semigroups
F 35.2. FreeGroup
F 35.2. FreeGroup
F 35.2. FreeGroup
F 35.2. FreeGroup
F 35.2. FreeGroup
F 35.2. IsFreeGroup
F 35.2. FreeMonoid!with example
F 35.2. FreeMonoid!with example
F 35.2. FreeMonoid!with example
F 35.2. FreeMonoid!with example
F 35.2. FreeMonoid!with example
F 35.2. FreeSemigroup
F 35.2. FreeSemigroup
F 35.2. FreeSemigroup
F 35.2. FreeSemigroup
F 35.2. FreeSemigroup
F 35.2. AssignGeneratorVariables
S 35.3. Comparison of Associative Words
F 35.3. equality!associative words
F 35.3. smaller!associative words
F 35.3. IsShortLexLessThanOrEqual
F 35.3. IsBasicWreathLessThanOrEqual
S 35.4. Operations for Associative Words
I 35.4. product!of words
I 35.4. quotient!of words
I 35.4. power!of words
I 35.4. conjugate!of a word
I 35.4. Comm!for words
I 35.4. LeftQuotient!for words
I 35.4. length!of a word
F 35.4. Length!of an associative word
F 35.4. ExponentSumWord
F 35.4. Subword
F 35.4. PositionWord
F 35.4. SubstitutedWord
F 35.4. SubstitutedWord
F 35.4. EliminatedWord
S 35.5. Operations for Associative Words by their Syllables
F 35.5. NumberSyllables
F 35.5. ExponentSyllable
F 35.5. GeneratorSyllable
F 35.5. SubSyllables
S 35.6. Representations for Associative Words
F 35.6. IsLetterAssocWordRep
F 35.6. IsLetterWordsFamily
F 35.6. IsBLetterAssocWordRep
F 35.6. IsWLetterAssocWordRep
F 35.6. IsBLetterWordsFamily
F 35.6. IsWLetterWordsFamily
F 35.6. IsSyllableAssocWordRep
F 35.6. IsSyllableWordsFamily
F 35.6. Is8BitsFamily
F 35.6. Is16BitsFamily
F 35.6. Is32BitsFamily
F 35.6. IsInfBitsFamily
F 35.6. LetterRepAssocWord
F 35.6. LetterRepAssocWord
F 35.6. AssocWordByLetterRep
S 35.7. The External Representation for Associative Words
S 35.8. Straight Line Programs
F 35.8. IsStraightLineProgram
F 35.8. StraightLineProgram
F 35.8. StraightLineProgram
F 35.8. StraightLineProgramNC
F 35.8. StraightLineProgramNC
F 35.8. LinesOfStraightLineProgram
F 35.8. NrInputsOfStraightLineProgram
F 35.8. ResultOfStraightLineProgram
F 35.8. StringOfResultOfStraightLineProgram
F 35.8. CompositionOfStraightLinePrograms
F 35.8. IntegratedStraightLineProgram
S 35.9. Straight Line Program Elements
F 35.9. IsStraightLineProgElm
F 35.9. StraightLineProgElm
F 35.9. StraightLineProgGens
F 35.9. EvalStraightLineProgElm
F 35.9. StretchImportantSLPElement
C rws.tex 36. Rewriting Systems
S 36.1. Operations on rewriting systems
F 36.1. IsRewritingSystem
F 36.1. Rules
F 36.1. OrderOfRewritingSystem
F 36.1. OrderingOfRewritingSystem
F 36.1. ReducedForm
F 36.1. IsConfluent
F 36.1. IsConfluent
F 36.1. ConfluentRws
F 36.1. IsReduced
F 36.1. ReduceRules
F 36.1. AddRule
F 36.1. AddRuleReduced
F 36.1. MakeConfluent
F 36.1. GeneratorsOfRws
F 36.1. AddGenerators
S 36.2. Operations on elements of the algebra
F 36.2. ReducedProduct
F 36.2. ReducedSum
F 36.2. ReducedOne
F 36.2. ReducedAdditiveInverse
F 36.2. ReducedComm
F 36.2. ReducedConjugate
F 36.2. ReducedDifference
F 36.2. ReducedInverse
F 36.2. ReducedLeftQuotient
F 36.2. ReducedPower
F 36.2. ReducedQuotient
F 36.2. ReducedScalarProduct
F 36.2. ReducedZero
S 36.3. Properties of rewriting systems
F 36.3. IsBuiltFromAdditiveMagmaWithInverses
F 36.3. IsBuiltFromMagma
F 36.3. IsBuiltFromMagmaWithOne
F 36.3. IsBuiltFromMagmaWithInverses
F 36.3. IsBuiltFromSemigroup
F 36.3. IsBuiltFromGroup
S 36.4. Rewriting in Groups and Monoids
S 36.5. Developing rewriting systems
C groups.tex 37. Groups
S 37.1. Group Elements
I 37.1. order! of a group
S 37.2. Creating Groups
F 37.2. Group
F 37.2. Group
F 37.2. Group
F 37.2. GroupWithGenerators
F 37.2. GroupWithGenerators
F 37.2. GeneratorsOfGroup
F 37.2. AsGroup
F 37.2. ConjugateGroup
F 37.2. IsGroup
F 37.2. InfoGroup
S 37.3. Subgroups
F 37.3. Subgroup
F 37.3. SubgroupNC
F 37.3. Index
F 37.3. IndexNC
F 37.3. IndexInWholeGroup
F 37.3. AsSubgroup
F 37.3. IsSubgroup
F 37.3. IsNormal
F 37.3. IsCharacteristicSubgroup
F 37.3. ConjugateSubgroup
F 37.3. ConjugateSubgroups
F 37.3. IsSubnormal
F 37.3. SubgroupByProperty
F 37.3. SubgroupShell
S 37.4. Closures of (Sub)groups
F 37.4. ClosureGroup
F 37.4. ClosureGroupAddElm
F 37.4. ClosureGroupCompare
F 37.4. ClosureGroupIntest
F 37.4. ClosureGroupDefault
F 37.4. ClosureSubgroup
F 37.4. ClosureSubgroupNC
S 37.5. Expressing Group Elements as Words in Generators
I 37.5. factorization
I 37.5. words!in generators
F 37.5. Factorization
S 37.6. Cosets
I 37.6. right cosets
I 37.6. coset
F 37.6. RightCoset
F 37.6. RightCosets
F 37.6. RightCosetsNC
F 37.6. CanonicalRightCosetElement
F 37.6. IsRightCoset
I 37.6. left cosets
S 37.7. Transversals
F 37.7. RightTransversal
S 37.8. Double Cosets
F 37.8. DoubleCoset
F 37.8. RepresentativesContainedRightCosets
F 37.8. DoubleCosets!operation
F 37.8. DoubleCosetsNC!operation
F 37.8. IsDoubleCoset
F 37.8. DoubleCosetRepsAndSizes
F 37.8. InfoCoset
S 37.9. Conjugacy Classes
F 37.9. ConjugacyClass
F 37.9. ConjugacyClasses!attribute
F 37.9. ConjugacyClassesByRandomSearch
F 37.9. ConjugacyClassesByOrbits
F 37.9. NrConjugacyClasses
F 37.9. RationalClass
F 37.9. RationalClasses
F 37.9. GaloisGroup!of rational class of a group
F 37.9. IsConjugate
F 37.9. IsConjugate
S 37.10. Normal Structure
I 37.10. normalizer
F 37.10. Normalizer
F 37.10. Normalizer
F 37.10. Core
F 37.10. PCore
I 37.10. O_p(G)!see PCore
F 37.10. NormalClosure
F 37.10. NormalIntersection
F 37.10. Complementclasses
F 37.10. InfoComplement
S 37.11. Specific and Parametrized Subgroups
F 37.11. TrivialSubgroup
F 37.11. CommutatorSubgroup
F 37.11. DerivedSubgroup
F 37.11. CommutatorLength
F 37.11. FittingSubgroup
F 37.11. FrattiniSubgroup
F 37.11. PrefrattiniSubgroup
F 37.11. PerfectResiduum
F 37.11. RadicalGroup
F 37.11. Socle
F 37.11. SupersolvableResiduum
F 37.11. PRump
S 37.12. Sylow Subgroups and Hall Subgroups
F 37.12. SylowSubgroup
F 37.12. SylowComplement
F 37.12. HallSubgroup
F 37.12. SylowSystem
F 37.12. ComplementSystem
F 37.12. HallSystem
S 37.13. Subgroups characterized by prime powers
F 37.13. Omega
F 37.13. Agemo
S 37.14. Group Properties
F 37.14. IsCyclic
F 37.14. IsElementaryAbelian
F 37.14. IsNilpotentGroup
F 37.14. NilpotencyClassOfGroup
F 37.14. IsPerfectGroup
F 37.14. IsSolvableGroup
F 37.14. IsPolycyclicGroup
F 37.14. IsSupersolvableGroup
F 37.14. IsMonomialGroup
F 37.14. IsSimpleGroup
F 37.14. IsomorphismTypeInfoFiniteSimpleGroup
F 37.14. IsFinitelyGeneratedGroup
F 37.14. IsSubsetLocallyFiniteGroup
I 37.14. p-group
F 37.14. IsPGroup
F 37.14. PrimePGroup
F 37.14. PClassPGroup
F 37.14. RankPGroup
F 37.14. IsPSolvable
F 37.14. IsPNilpotent
S 37.15. Numerical Group Attributes
F 37.15. AbelianInvariants!for groups
F 37.15. Exponent
F 37.15. EulerianFunction
S 37.16. Subgroup Series
F 37.16. ChiefSeries
F 37.16. ChiefSeriesThrough
F 37.16. ChiefSeriesUnderAction
F 37.16. SubnormalSeries
F 37.16. CompositionSeries
F 37.16. DisplayCompositionSeries
F 37.16. DerivedSeriesOfGroup
F 37.16. DerivedLength
F 37.16. ElementaryAbelianSeries
F 37.16. ElementaryAbelianSeriesLargeSteps
F 37.16. ElementaryAbelianSeries
F 37.16. InvariantElementaryAbelianSeries
F 37.16. LowerCentralSeriesOfGroup
F 37.16. UpperCentralSeriesOfGroup
F 37.16. PCentralSeries
F 37.16. JenningsSeries
F 37.16. DimensionsLoewyFactors
F 37.16. AscendingChain
F 37.16. IntermediateGroup
F 37.16. IntermediateSubgroups
S 37.17. Factor Groups
F 37.17. NaturalHomomorphismByNormalSubgroup
F 37.17. NaturalHomomorphismByNormalSubgroupNC
F 37.17. FactorGroup
F 37.17. FactorGroupNC
F 37.17. CommutatorFactorGroup
F 37.17. MaximalAbelianQuotient
F 37.17. HasAbelianFactorGroup
F 37.17. HasElementaryAbelianFactorGroup
F 37.17. CentralizerModulo
S 37.18. Sets of Subgroups
F 37.18. ConjugacyClassSubgroups
F 37.18. IsConjugacyClassSubgroupsRep
F 37.18. IsConjugacyClassSubgroupsByStabilizerRep
F 37.18. ConjugacyClassesSubgroups
F 37.18. ConjugacyClassesMaximalSubgroups
F 37.18. MaximalSubgroupClassReps
F 37.18. MaximalSubgroups
F 37.18. NormalSubgroups
F 37.18. MaximalNormalSubgroups
F 37.18. MinimalNormalSubgroups
S 37.19. Subgroup Lattice
F 37.19. LatticeSubgroups
F 37.19. ClassElementLattice
F 37.19. MaximalSubgroupsLattice
F 37.19. MinimalSupergroupsLattice
F 37.19. RepresentativesPerfectSubgroups
F 37.19. RepresentativesSimpleSubgroups
F 37.19. ConjugacyClassesPerfectSubgroups
F 37.19. Zuppos
F 37.19. InfoLattice
S 37.20. Specific Methods for Subgroup Lattice Computations
F 37.20. LatticeByCyclicExtension
F 37.20. InvariantSubgroupsElementaryAbelianGroup
F 37.20. SubgroupsSolvableGroup
F 37.20. SizeConsiderFunction
F 37.20. ExactSizeConsiderFunction
F 37.20. InfoPcSubgroup
S 37.21. Special Generating Sets
F 37.21. GeneratorsSmallest
F 37.21. LargestElementGroup
F 37.21. MinimalGeneratingSet
F 37.21. SmallGeneratingSet
F 37.21. IndependentGeneratorsOfAbelianGroup
S 37.22. 1-Cohomology
I 37.22. one cohomology
I 37.22. cohomology
I 37.22. cocycles
F 37.22. OneCocycles
F 37.22. OneCocycles
F 37.22. OneCocycles
F 37.22. OneCocycles
F 37.22. OneCoboundaries
F 37.22. OCOneCocycles
F 37.22. ComplementclassesEA
F 37.22. InfoCoh
S 37.23. Schur Covers and Multipliers
I 37.23. Darstellungsgruppe!see EpimorphismSchurCover
F 37.23. EpimorphismSchurCover
F 37.23. SchurCover
F 37.23. AbelianInvariantsMultiplier
I 37.23. Multiplier
I 37.23. Schur multiplier
S 37.24. Tests for the Availability of Methods
F 37.24. CanEasilyTestMembership
F 37.24. CanComputeSize
F 37.24. CanComputeSizeAnySubgroup
F 37.24. CanComputeIndex
F 37.24. CanComputeIsSubset
F 37.24. KnowsHowToDecompose
F 37.24. KnowsHowToDecompose
C grphomom.tex 38. Group Homomorphisms
S 38.1. Creating Group Homomorphisms
F 38.1. GroupHomomorphismByImages
F 38.1. GroupHomomorphismByImagesNC
F 38.1. GroupGeneralMappingByImages
F 38.1. GroupHomomorphismByFunction
F 38.1. GroupHomomorphismByFunction
F 38.1. GroupHomomorphismByFunction
F 38.1. AsGroupGeneralMappingByImages
S 38.2. Operations for Group Homomorphisms
I 38.2. Inverse!group homomorphism
S 38.3. Efficiency of Homomorphisms
F 38.3. ImagesSmallestGenerators
S 38.4. Homomorphism for very large groups
S 38.5. Nice Monomorphisms
F 38.5. IsHandledByNiceMonomorphism
F 38.5. NiceMonomorphism
F 38.5. NiceObject
F 38.5. IsCanonicalNiceMonomorphism
S 38.6. Group Automorphisms
F 38.6. ConjugatorIsomorphism
F 38.6. ConjugatorAutomorphism
F 38.6. ConjugatorAutomorphismNC
F 38.6. InnerAutomorphism
F 38.6. InnerAutomorphismNC
F 38.6. IsConjugatorIsomorphism
F 38.6. IsConjugatorAutomorphism
F 38.6. IsInnerAutomorphism
F 38.6. ConjugatorOfConjugatorIsomorphism
S 38.7. Groups of Automorphisms
F 38.7. IsGroupOfAutomorphisms
F 38.7. AutomorphismDomain
F 38.7. AutomorphismGroup
F 38.7. IsAutomorphismGroup
F 38.7. InnerAutomorphismsAutomorphismGroup
F 38.7. InducedAutomorphism
S 38.8. Calculating with Group Automorphisms
F 38.8. AssignNiceMonomorphismAutomorphismGroup
F 38.8. NiceMonomorphismAutomGroup
S 38.9. Searching for Homomorphisms
I 38.9. homomorphisms!find all
F 38.9. IsomorphismGroups
I 38.9. isomorphisms!find all
F 38.9. GQuotients
I 38.9. epimorphisms!find all
I 38.9. projections!find all
F 38.9. IsomorphicSubgroups
I 38.9. embeddings!find all
I 38.9. monomorphisms!find all
F 38.9. MorClassLoop
S 38.10. Representations for Group Homomorphisms
F 38.10. IsGroupGeneralMappingByImages
F 38.10. IsGroupGeneralMappingByAsGroupGeneralMappingByImages
F 38.10. IsPreimagesByAsGroupGeneralMappingByImages
F 38.10. IsPermGroupGeneralMappingByImages
F 38.10. IsPermGroupHomomorphismByImages
F 38.10. IsToPermGroupGeneralMappingByImages
F 38.10. IsToPermGroupHomomorphismByImages
F 38.10. IsGroupGeneralMappingByPcgs
F 38.10. IsPcGroupGeneralMappingByImages
F 38.10. IsPcGroupHomomorphismByImages
F 38.10. IsToPcGroupGeneralMappingByImages
F 38.10. IsToPcGroupHomomorphismByImages
F 38.10. IsFromFpGroupGeneralMappingByImages
F 38.10. IsFromFpGroupHomomorphismByImages
F 38.10. IsFromFpGroupStdGensGeneralMappingByImages
F 38.10. IsFromFpGroupStdGensHomomorphismByImages
C grpoper.tex 39. Group Actions
I 39.0. group actions
I 39.0. G-sets
S 39.1. About Group Actions
I 39.1. group actions!operations syntax
S 39.2. Basic Actions
I 39.2. group actions
I 39.2. actions
I 39.2. group operations
F 39.2. OnPoints
I 39.2. conjugation
I 39.2. action!by conjugation
F 39.2. OnRight
F 39.2. OnLeftInverse
F 39.2. OnSets
I 39.2. action!on sets
I 39.2. action!on blocks
F 39.2. OnTuples
F 39.2. OnPairs
F 39.2. OnSetsSets
F 39.2. OnSetsDisjointSets
F 39.2. OnSetsTuples
F 39.2. OnTuplesSets
F 39.2. OnTuplesTuples
F 39.2. OnLines
F 39.2. OnIndeterminates!as a permutation action
F 39.2. Permuted!as a permutation action
F 39.2. OnSubspacesByCanonicalBasis
S 39.3. Orbits
F 39.3. Orbit
F 39.3. Orbits!operation/attribute
F 39.3. Orbits!operation/attribute
F 39.3. OrbitsDomain
F 39.3. OrbitsDomain
F 39.3. OrbitLength
F 39.3. OrbitLengths
F 39.3. OrbitLengths
F 39.3. OrbitLengthsDomain
F 39.3. OrbitLengthsDomain
S 39.4. Stabilizers
I 39.4. point stabilizer
I 39.4. set stabilizer
I 39.4. tuple stabilizer
F 39.4. OrbitStabilizer
F 39.4. Stabilizer
F 39.4. OrbitStabilizerAlgorithm
S 39.5. Elements with Prescribed Images
I 39.5. transporter
F 39.5. RepresentativeAction
S 39.6. The Permutation Image of an Action
F 39.6. ActionHomomorphism
F 39.6. ActionHomomorphism
F 39.6. ActionHomomorphism
F 39.6. Action
F 39.6. Action
I 39.6. regular action
F 39.6. SparseActionHomomorphism
F 39.6. SortedSparseActionHomomorphism
S 39.7. Action of a group on itself
F 39.7. FactorCosetAction
F 39.7. RegularActionHomomorphism
F 39.7. AbelianSubfactorAction
S 39.8. Permutations Induced by Elements and Cycles
F 39.8. Permutation
F 39.8. Permutation
F 39.8. PermutationCycle
F 39.8. Cycle
F 39.8. CycleLength
F 39.8. Cycles
F 39.8. CycleLengths
S 39.9. Tests for Actions
F 39.9. IsTransitive!for group actions
F 39.9. IsTransitive!for group actions
I 39.9. transitive
F 39.9. Transitivity!for group actions
F 39.9. Transitivity!for group actions
F 39.9. RankAction
F 39.9. RankAction
F 39.9. IsSemiRegular
F 39.9. IsSemiRegular
I 39.9. semiregular
F 39.9. IsRegular
F 39.9. IsRegular
I 39.9. regular
F 39.9. Earns
F 39.9. Earns
F 39.9. IsPrimitive
F 39.9. IsPrimitive
I 39.9. primitive
S 39.10. Block Systems
F 39.10. Blocks
F 39.10. Blocks
F 39.10. MaximalBlocks
F 39.10. MaximalBlocks
F 39.10. RepresentativesMinimalBlocks
F 39.10. RepresentativesMinimalBlocks
F 39.10. AllBlocks
S 39.11. External Sets
I 39.11. G-sets
F 39.11. IsExternalSet
F 39.11. ExternalSet
F 39.11. ActingDomain
F 39.11. FunctionAction
F 39.11. HomeEnumerator
F 39.11. IsExternalSubset
F 39.11. ExternalSubset
F 39.11. IsExternalOrbit
F 39.11. ExternalOrbit
F 39.11. StabilizerOfExternalSet
F 39.11. ExternalOrbits
F 39.11. ExternalOrbits
F 39.11. ExternalOrbitsStabilizers
F 39.11. ExternalOrbitsStabilizers
F 39.11. CanonicalRepresentativeOfExternalSet
F 39.11. CanonicalRepresentativeDeterminatorOfExternalSet
F 39.11. ActorOfExternalSet
F 39.11. UnderlyingExternalSet
F 39.11. SurjectiveActionHomomorphismAttr
C permutat.tex 40. Permutations
F 40.0. IsPerm
F 40.0. IsPermCollection
F 40.0. IsPermCollColl
F 40.0. PermutationsFamily
S 40.1. Comparison of Permutations
F 40.1. equality test!for permutations
F 40.1. precedence test!for permutations
F 40.1. SmallestGeneratorPerm
S 40.2. Moved Points of Permutations
F 40.2. SmallestMovedPoint
F 40.2. SmallestMovedPoint
F 40.2. LargestMovedPoint
F 40.2. LargestMovedPoint
F 40.2. MovedPoints
F 40.2. MovedPoints
F 40.2. NrMovedPoints
F 40.2. NrMovedPoints
S 40.3. Sign and Cycle Structure
F 40.3. SignPerm
F 40.3. CycleStructurePerm
S 40.4. Creating Permutations
F 40.4. ListPerm
F 40.4. PermList
F 40.4. MappingPermListList
F 40.4. RestrictedPerm
C grpperm.tex 41. Permutation Groups
F 41.0. IsPermGroup
S 41.1. The Natural Action
F 41.1. OrbitPerms
F 41.1. OrbitsPerms
S 41.2. Computing a Permutation Representation
F 41.2. IsomorphismPermGroup
F 41.2. SmallerDegreePermutationRepresentation
S 41.3. Symmetric and Alternating Groups
F 41.3. IsNaturalSymmetricGroup
F 41.3. IsNaturalAlternatingGroup
F 41.3. IsSymmetricGroup
F 41.3. IsAlternatingGroup
F 41.3. SymmetricParentGroup
S 41.4. Primitive Groups
F 41.4. ONanScottType
F 41.4. SocleTypePrimitiveGroup
S 41.5. Stabilizer Chains
S 41.6. Randomized Methods for Permutation Groups
I 41.6. Schreier-Sims!random
S 41.7. Construction of Stabilizer Chains
F 41.7. StabChain
F 41.7. StabChain
F 41.7. StabChainOp
F 41.7. StabChainMutable
F 41.7. StabChainMutable
F 41.7. StabChainImmutable
F 41.7. StabChainOptions
F 41.7. DefaultStabChainOptions
F 41.7. StabChainBaseStrongGenerators
F 41.7. MinimalStabChain
S 41.8. Stabilizer Chain Records
S 41.9. Operations for Stabilizer Chains
F 41.9. BaseStabChain
F 41.9. BaseOfGroup
F 41.9. SizeStabChain
F 41.9. StrongGeneratorsStabChain
F 41.9. GroupStabChain
F 41.9. OrbitStabChain
F 41.9. IndicesStabChain
F 41.9. ListStabChain
F 41.9. ElementsStabChain
F 41.9. InverseRepresentative
F 41.9. SiftedPermutation
F 41.9. MinimalElementCosetStabChain
F 41.9. LargestElementStabChain
F 41.9. ApproximateSuborbitsStabilizerPermGroup
S 41.10. Low Level Routines to Modify and Create Stabilizer Chains
F 41.10. CopyStabChain
F 41.10. CopyOptionsDefaults
F 41.10. ChangeStabChain
F 41.10. ExtendStabChain
F 41.10. ReduceStabChain
F 41.10. RemoveStabChain
F 41.10. EmptyStabChain
F 41.10. InsertTrivialStabilizer
F 41.10. IsFixedStabilizer
F 41.10. AddGeneratorsExtendSchreierTree
S 41.11. Backtrack
F 41.11. SubgroupProperty
F 41.11. ElementProperty
F 41.11. TwoClosure
F 41.11. InfoBckt
S 41.12. Working with large degree permutation groups
C grpmat.tex 42. Matrix Groups
F 42.0. IsMatrixGroup
S 42.1. Attributes and Properties for Matrix Groups
F 42.1. DimensionOfMatrixGroup
F 42.1. DefaultFieldOfMatrixGroup
F 42.1. FieldOfMatrixGroup
F 42.1. TransposedMatrixGroup
S 42.2. Actions of Matrix Groups
F 42.2. ProjectiveActionOnFullSpace
F 42.2. ProjectiveActionHomomorphismMatrixGroup
F 42.2. BlowUpIsomorphism
S 42.3. GL and SL
F 42.3. IsGeneralLinearGroup
F 42.3. IsGL
F 42.3. IsNaturalGL
F 42.3. IsSpecialLinearGroup
F 42.3. IsSL
F 42.3. IsNaturalSL
F 42.3. IsSubgroupSL
S 42.4. Invariant Forms
F 42.4. InvariantBilinearForm
F 42.4. IsFullSubgroupGLorSLRespectingBilinearForm
F 42.4. InvariantSesquilinearForm
F 42.4. IsFullSubgroupGLorSLRespectingSesquilinearForm
F 42.4. InvariantQuadraticForm
F 42.4. IsFullSubgroupGLorSLRespectingQuadraticForm
S 42.5. Matrix Groups in Characteristic 0
F 42.5. IsCyclotomicMatrixGroup
F 42.5. IsRationalMatrixGroup
F 42.5. IsIntegerMatrixGroup
F 42.5. IsNaturalGLnZ
F 42.5. IsNaturalSLnZ
F 42.5. InvariantLattice
F 42.5. NormalizerInGLnZ
F 42.5. CentralizerInGLnZ
F 42.5. ZClassRepsQClass
F 42.5. IsBravaisGroup
F 42.5. BravaisGroup
F 42.5. BravaisSubgroups
F 42.5. BravaisSupergroups
F 42.5. NormalizerInGLnZBravaisGroup
S 42.6. Acting OnRight and OnLeft
F 42.6. CrystGroupDefaultAction
F 42.6. SetCrystGroupDefaultAction
C pcgs.tex 43. Polycyclic Groups
S 43.1. Polycyclic Generating Systems
S 43.2. Computing a Pcgs
F 43.2. Pcgs
F 43.2. IsPcgs
F 43.2. CanEasilyComputePcgs
S 43.3. Defining a Pcgs Yourself
F 43.3. PcgsByPcSequence
F 43.3. PcgsByPcSequenceNC
S 43.4. Elementary Operations for a Pcgs
F 43.4. RelativeOrders!of a pcgs
F 43.4. IsFiniteOrdersPcgs
F 43.4. IsPrimeOrdersPcgs
F 43.4. PcSeries
F 43.4. GroupOfPcgs
F 43.4. OneOfPcgs
S 43.5. Elementary Operations for a Pcgs and an Element
F 43.5. RelativeOrderOfPcElement
F 43.5. ExponentOfPcElement
F 43.5. ExponentsOfPcElement
F 43.5. ExponentsOfPcElement
F 43.5. DepthOfPcElement
F 43.5. LeadingExponentOfPcElement
F 43.5. PcElementByExponents
F 43.5. PcElementByExponentsNC
F 43.5. PcElementByExponentsNC
F 43.5. LinearCombinationPcgs
F 43.5. SiftedPcElement
F 43.5. CanonicalPcElement
F 43.5. ReducedPcElement
F 43.5. CleanedTailPcElement
F 43.5. HeadPcElementByNumber
S 43.6. Exponents of Special Products
F 43.6. ExponentsConjugateLayer
F 43.6. ExponentsOfRelativePower
F 43.6. ExponentsOfConjugate
F 43.6. ExponentsOfCommutator
S 43.7. Subgroups of Polycyclic Groups - Induced Pcgs
F 43.7. IsInducedPcgs
F 43.7. InducedPcgsByPcSequence
F 43.7. InducedPcgsByPcSequenceNC
F 43.7. InducedPcgsByPcSequenceNC
F 43.7. ParentPcgs
F 43.7. InducedPcgs
F 43.7. InducedPcgsByGenerators
F 43.7. InducedPcgsByGeneratorsNC
F 43.7. InducedPcgsByPcSequenceAndGenerators
F 43.7. LeadCoeffsIGS
F 43.7. ExtendedPcgs
F 43.7. SubgroupByPcgs
S 43.8. Subgroups of Polycyclic Groups - Canonical Pcgs
F 43.8. IsCanonicalPcgs
F 43.8. CanonicalPcgs
S 43.9. Factor Groups of Polycyclic Groups - Modulo Pcgs
F 43.9. ModuloPcgs
F 43.9. IsModuloPcgs
F 43.9. NumeratorOfModuloPcgs
F 43.9. DenominatorOfModuloPcgs
F 43.9. modulo!for pcgs
F 43.9. CorrespondingGeneratorsByModuloPcgs
F 43.9. CanonicalPcgsByGeneratorsWithImages
S 43.10. Factor Groups of Polycyclic Groups in their Own Representation
F 43.10. ProjectedPcElement
F 43.10. ProjectedInducedPcgs
F 43.10. LiftedPcElement
F 43.10. LiftedInducedPcgs
S 43.11. Pcgs and Normal Series
F 43.11. IsPcgsElementaryAbelianSeries
F 43.11. PcgsElementaryAbelianSeries
F 43.11. PcgsElementaryAbelianSeries
F 43.11. IndicesEANormalSteps
F 43.11. EANormalSeriesByPcgs
F 43.11. IsPcgsCentralSeries
F 43.11. PcgsCentralSeries
F 43.11. IndicesCentralNormalSteps
F 43.11. CentralNormalSeriesByPcgs
F 43.11. IsPcgsPCentralSeriesPGroup
F 43.11. PcgsPCentralSeriesPGroup
F 43.11. IndicesPCentralNormalStepsPGroup
F 43.11. PCentralNormalSeriesByPcgsPGroup
F 43.11. IsPcgsChiefSeries
F 43.11. PcgsChiefSeries
F 43.11. IndicesChiefNormalSteps
F 43.11. ChiefNormalSeriesByPcgs
F 43.11. IndicesNormalSteps
F 43.11. NormalSeriesByPcgs
S 43.12. Sum and Intersection of Pcgs
F 43.12. SumFactorizationFunctionPcgs
S 43.13. Special Pcgs
F 43.13. IsSpecialPcgs
F 43.13. SpecialPcgs!attribute
F 43.13. SpecialPcgs!attribute
F 43.13. LGWeights
F 43.13. LGLayers
F 43.13. LGFirst
F 43.13. LGLength
F 43.13. IsInducedPcgsWrtSpecialPcgs
F 43.13. InducedPcgsWrtSpecialPcgs
S 43.14. Action on Subfactors Defined by a Pcgs
F 43.14. VectorSpaceByPcgsOfElementaryAbelianGroup
F 43.14. LinearOperation
F 43.14. LinearAction
F 43.14. LinearOperationLayer
F 43.14. LinearActionLayer
F 43.14. AffineOperation
F 43.14. AffineAction
F 43.14. AffineOperationLayer
F 43.14. AffineActionLayer
S 43.15. Orbit Stabilizer Methods for Polycyclic Groups
F 43.15. StabilizerPcgs
F 43.15. Pcgs_OrbitStabilizer
S 43.16. Operations which have Special Methods for Groups with Pcgs
I 43.16. IsNilpotent!for groups with pcgs
I 43.16. IsSupersolvable!for groups with pcgs
I 43.16. Size!for groups with pcgs
I 43.16. CompositionSeries!for groups with pcgs
I 43.16. ConjugacyClasses!for groups with pcgs
I 43.16. Centralizer!for groups with pcgs
I 43.16. FrattiniSubgroup!for groups with pcgs
I 43.16. PrefrattiniSubgroup!for groups with pcgs
I 43.16. MaximalSubgroups!for groups with pcgs
I 43.16. HallSystem!for groups with pcgs
I 43.16. MinimalGeneratingSet!for groups with pcgs
I 43.16. Centre!for groups with pcgs
I 43.16. Intersection!for groups with pcgs
I 43.16. AutomorphismGroup!for groups with pcgs
I 43.16. IrreducibleModules!for groups with pcgs
S 43.17. Conjugacy Classes in Solvable Groups
F 43.17. ClassesSolvableGroup
F 43.17. CentralizerSizeLimitConsiderFunction
C grppc.tex 44. Pc Groups
S 44.1. The family pcgs
F 44.1. FamilyPcgs
F 44.1. IsFamilyPcgs
F 44.1. InducedPcgsWrtFamilyPcgs
F 44.1. IsParentPcgsFamilyPcgs
S 44.2. Elements of pc groups
F 44.2. equality!pcwords
F 44.2. smaller!pcwords
S 44.3. Pc groups versus fp groups
F 44.3. IsPcGroup
F 44.3. IsomorphismFpGroupByPcgs
S 44.4. Constructing Pc Groups
F 44.4. PcGroupFpGroup
F 44.4. SingleCollector
F 44.4. CombinatorialCollector
F 44.4. SetConjugate
F 44.4. SetCommutator
F 44.4. SetPower
F 44.4. GroupByRws
F 44.4. GroupByRwsNC
F 44.4. IsConfluent!for pc groups
F 44.4. IsomorphismRefinedPcGroup
I 44.4. isomorphic!pc group
F 44.4. RefinedPcGroup
S 44.5. Computing Pc Groups
F 44.5. PcGroupWithPcgs
F 44.5. IsomorphismPcGroup
I 44.5. isomorphic!pc group
F 44.5. IsomorphismSpecialPcGroup
S 44.6. Saving a Pc Group
F 44.6. GapInputPcGroup
S 44.7. Operations for Pc Groups
S 44.8. 2-Cohomology and Extensions
F 44.8. TwoCoboundaries
F 44.8. TwoCocycles
F 44.8. TwoCohomology
F 44.8. Extensions
F 44.8. Extension
F 44.8. ExtensionNC
F 44.8. SplitExtension
F 44.8. ModuleOfExtension
F 44.8. CompatiblePairs
F 44.8. ExtensionRepresentatives
F 44.8. SplitExtensions
S 44.9. Coding a Pc Presentation
F 44.9. CodePcgs
F 44.9. CodePcGroup
F 44.9. PcGroupCode
F 44.9. PcGroupCodeRec
S 44.10. Random Isomorphism Testing
F 44.10. RandomIsomorphismTest
C grpfp.tex 45. Finitely Presented Groups
F 45.0. IsSubgroupFpGroup
F 45.0. IsFpGroup
F 45.0. InfoFpGroup
S 45.1. Creating Finitely Presented Groups
F 45.1. quotient!for finitely presented groups
F 45.1. FactorGroupFpGroupByRels
S 45.2. Comparison of Elements of Finitely Presented Groups
F 45.2. equality!elements of finitely presented groups
F 45.2. smaller!elements of finitely presented groups
F 45.2. FpElmComparisonMethod
F 45.2. SetReducedMultiplication
F 45.2. SetReducedMultiplication
F 45.2. SetReducedMultiplication
S 45.3. Preimages in the Free Group
F 45.3. FreeGroupOfFpGroup
F 45.3. FreeGeneratorsOfFpGroup
F 45.3. FreeGeneratorsOfWholeGroup
F 45.3. RelatorsOfFpGroup
F 45.3. UnderlyingElement!fp group elements
F 45.3. ElementOfFpGroup
S 45.4. Operations for Finitely Presented Groups
S 45.5. Coset Tables and Coset Enumeration
F 45.5. CosetTable
F 45.5. TracedCosetFpGroup
F 45.5. FactorCosetAction!for fp groups
F 45.5. FactorCosetOperation
F 45.5. CosetTableBySubgroup
F 45.5. CosetTableFromGensAndRels
F 45.5. CosetTableDefaultMaxLimit
F 45.5. CosetTableDefaultLimit
F 45.5. MostFrequentGeneratorFpGroup
F 45.5. IndicesInvolutaryGenerators
S 45.6. Standardization of coset tables
F 45.6. CosetTableStandard
F 45.6. StandardizeTable
S 45.7. Coset tables for subgroups in the whole group
F 45.7. CosetTableInWholeGroup
F 45.7. TryCosetTableInWholeGroup
F 45.7. SubgroupOfWholeGroupByCosetTable
S 45.8. Augmented Coset Tables and Rewriting
F 45.8. AugmentedCosetTableInWholeGroup
F 45.8. AugmentedCosetTableMtc
F 45.8. AugmentedCosetTableRrs
F 45.8. RewriteWord
S 45.9. Low Index Subgroups
F 45.9. LowIndexSubgroupsFpGroup
S 45.10. Converting Groups to Finitely Presented Groups
F 45.10. IsomorphismFpGroup
F 45.10. IsomorphismFpGroupByGenerators
F 45.10. IsomorphismFpGroupByGeneratorsNC
S 45.11. New Presentations and Presentations for Subgroups
I 45.11. IsomorphismFpGroup!for subgroups of fp groups
F 45.11. IsomorphismSimplifiedFpGroup
S 45.12. Preimages under Homomorphisms from an FpGroup
F 45.12. SubgroupOfWholeGroupByQuotientSubgroup
F 45.12. IsSubgroupOfWholeGroupByQuotientRep
F 45.12. AsSubgroupOfWholeGroupByQuotient
F 45.12. DefiningQuotientHomomorphism
S 45.13. Quotient Methods
F 45.13. PQuotient
F 45.13. EpimorphismQuotientSystem
F 45.13. EpimorphismPGroup
F 45.13. EpimorphismPGroup
F 45.13. EpimorphismNilpotentQuotient
S 45.14. Abelian Invariants for Subgroups
F 45.14. AbelianInvariantsSubgroupFpGroup
F 45.14. AbelianInvariantsSubgroupFpGroupMtc
F 45.14. AbelianInvariantsSubgroupFpGroupRrs
F 45.14. AbelianInvariantsSubgroupFpGroupRrs
F 45.14. AbelianInvariantsNormalClosureFpGroup
F 45.14. AbelianInvariantsNormalClosureFpGroupRrs
S 45.15. Testing Finiteness of Finitely Presented Groups
F 45.15. NewmanInfinityCriterion
C pres.tex 46. Presentations and Tietze Transformations
S 46.1. Creating Presentations
F 46.1. PresentationFpGroup
F 46.1. TzSort
F 46.1. GeneratorsOfPresentation
F 46.1. FpGroupPresentation
F 46.1. PresentationViaCosetTable
F 46.1. PresentationViaCosetTable
S 46.2. SimplifiedFpGroup
F 46.2. SimplifiedFpGroup
S 46.3. Subgroup Presentations
I 46.3. Schreier
F 46.3. PresentationSubgroup
F 46.3. PresentationSubgroupRrs
F 46.3. PresentationSubgroupRrs
F 46.3. PrimaryGeneratorWords
F 46.3. PresentationSubgroupMtc
F 46.3. PresentationNormalClosureRrs
F 46.3. PresentationNormalClosure
S 46.4. Relators in a Presentation
F 46.4. TietzeWordAbstractWord
F 46.4. AbstractWordTietzeWord
S 46.5. Printing Presentations
F 46.5. TzPrintGenerators
F 46.5. TzPrintRelators
F 46.5. TzPrintLengths
F 46.5. TzPrintStatus
F 46.5. TzPrintPresentation
F 46.5. TzPrint
F 46.5. TzPrintPairs
S 46.6. Changing Presentations
F 46.6. AddGenerator
F 46.6. TzNewGenerator
F 46.6. AddRelator
F 46.6. RemoveRelator
S 46.7. Tietze Transformations
F 46.7. TzGo
F 46.7. SimplifyPresentation
F 46.7. TzGoGo
S 46.8. Elementary Tietze Transformations
F 46.8. TzEliminate
F 46.8. TzEliminate
F 46.8. TzEliminate
F 46.8. TzSearch
F 46.8. TzSearchEqual
F 46.8. TzFindCyclicJoins
S 46.9. Tietze Transformations that introduce new Generators
F 46.9. TzSubstitute
F 46.9. TzSubstitute
F 46.9. TzSubstituteCyclicJoins
S 46.10. Tracing generator images through Tietze transformations
F 46.10. TzInitGeneratorImages
F 46.10. OldGeneratorsOfPresentation
F 46.10. TzImagesOldGens
F 46.10. TzPreImagesNewGens
F 46.10. TzPrintGeneratorImages
S 46.11. DecodeTree
F 46.11. DecodeTree
I 46.11. secondary subgroup generators
I 46.11. primary subgroup generators
I 46.11. subgroup generators tree
S 46.12. Tietze Options
F 46.12. TzOptions
F 46.12. TzPrintOptions
C grpprod.tex 47. Group Products
S 47.1. Direct Products
F 47.1. DirectProduct
F 47.1. DirectProductOp
I 47.1. Embedding!example for direct products
I 47.1. Projection!example for direct products
S 47.2. Semidirect Products
F 47.2. SemidirectProduct
F 47.2. SemidirectProduct
I 47.2. Embedding!example for semidirect products
I 47.2. Projection!example for semidirect products
S 47.3. Subdirect Products
F 47.3. SubdirectProduct
I 47.3. Projection!example for subdirect products
F 47.3. SubdirectProducts
S 47.4. Wreath Products
F 47.4. WreathProduct
F 47.4. WreathProduct
I 47.4. Embedding!example for wreath products
I 47.4. Projection!example for wreath products
F 47.4. WreathProductImprimitiveAction
F 47.4. WreathProductProductAction
F 47.4. KuKGenerators
I 47.4. Krasner-Kaloujnine theorem
I 47.4. Wreath product embedding
S 47.5. Embeddings and Projections for Group Products
F 47.5. Embedding!for group products
F 47.5. Projection!for group products
C grplib.tex 48. Group Libraries
S 48.1. Basic Groups
F 48.1. TrivialGroup
F 48.1. CyclicGroup
F 48.1. AbelianGroup
F 48.1. ElementaryAbelianGroup
F 48.1. DihedralGroup
F 48.1. ExtraspecialGroup
F 48.1. AlternatingGroup
F 48.1. AlternatingGroup
F 48.1. SymmetricGroup
F 48.1. SymmetricGroup
F 48.1. MathieuGroup
F 48.1. SuzukiGroup
F 48.1. Sz
F 48.1. ReeGroup
F 48.1. Ree
S 48.2. Classical Groups
F 48.2. GeneralLinearGroup
F 48.2. GL
F 48.2. GeneralLinearGroup
F 48.2. GL
F 48.2. SpecialLinearGroup
F 48.2. SL
F 48.2. SpecialLinearGroup
F 48.2. SL
I 48.2. OnLines!example
F 48.2. GeneralUnitaryGroup
F 48.2. GU
F 48.2. SpecialUnitaryGroup
F 48.2. SU
F 48.2. SymplecticGroup
F 48.2. Sp
F 48.2. SP
F 48.2. GeneralOrthogonalGroup
F 48.2. GO
F 48.2. SpecialOrthogonalGroup
F 48.2. SO
F 48.2. ProjectiveGeneralLinearGroup
F 48.2. PGL
F 48.2. ProjectiveSpecialLinearGroup
F 48.2. PSL
F 48.2. ProjectiveGeneralUnitaryGroup
F 48.2. PGU
F 48.2. ProjectiveSpecialUnitaryGroup
F 48.2. PSU
F 48.2. ProjectiveSymplecticGroup
F 48.2. PSP
F 48.2. PSp
S 48.3. Conjugacy Classes in Classical Groups
I 48.3. ConjugacyClasses!for linear groups
F 48.3. NrConjugacyClassesGL
F 48.3. NrConjugacyClassesGU
F 48.3. NrConjugacyClassesSL
F 48.3. NrConjugacyClassesSU
F 48.3. NrConjugacyClassesPGL
F 48.3. NrConjugacyClassesPGU
F 48.3. NrConjugacyClassesPSL
F 48.3. NrConjugacyClassesPSU
F 48.3. NrConjugacyClassesSLIsogeneous
F 48.3. NrConjugacyClassesSUIsogeneous
S 48.4. Constructors for Basic Groups
S 48.5. Selection Functions
I 48.5. AllPrimitiveGroups
I 48.5. AllTransitiveGroups
F 48.5. AllLibraryGroups
I 48.5. OnePrimitiveGroup
I 48.5. OneTransitiveGroup
F 48.5. OneLibraryGroup
S 48.6. Transitive Permutation Groups
F 48.6. TransitiveGroup
F 48.6. NrTransitiveGroups
F 48.6. TransitiveIdentification
S 48.7. Small Groups
I 48.7. TwoGroup library
I 48.7. ThreeGroup library
F 48.7. SmallGroup
F 48.7. SmallGroup
F 48.7. AllSmallGroups
F 48.7. OneSmallGroup
F 48.7. NumberSmallGroups
F 48.7. IdSmallGroup
F 48.7. IdGroup
F 48.7. IdsOfAllSmallGroups
F 48.7. IdGap3SolvableGroup
F 48.7. Gap3CatalogueIdGroup
F 48.7. SmallGroupsInformation
F 48.7. UnloadSmallGroupsData
S 48.8. Finite Perfect Groups
I 48.8. perfect groups
F 48.8. SizesPerfectGroups
F 48.8. PerfectGroup
F 48.8. PerfectGroup
F 48.8. PerfectIdentification
F 48.8. NumberPerfectGroups
F 48.8. NumberPerfectLibraryGroups
F 48.8. SizeNumbersPerfectGroups
F 48.8. DisplayInformationPerfectGroups
F 48.8. DisplayInformationPerfectGroups
F 48.8. DisplayInformationPerfectGroups
S 48.9. Primitive Permutation Groups
F 48.9. PrimitiveGroup
F 48.9. NrPrimitiveGroups
F 48.9. PrimitiveGroupsIterator
F 48.9. COHORTS_PRIMITIVE_GROUPS
S 48.10. Index numbers of primitive groups
F 48.10. PrimitiveIdentification
F 48.10. SimsNo
F 48.10. PRIMITIVE_INDICES_MAGMA
S 48.11. Irreducible Solvable Matrix Groups
F 48.11. IrreducibleSolvableGroupMS
F 48.11. NumberIrreducibleSolvableGroups
F 48.11. AllIrreducibleSolvableGroups
F 48.11. OneIrreducibleSolvableGroup
F 48.11. PrimitiveIndexIrreducibleSolvableGroup
F 48.11. IrreducibleSolvableGroup
S 48.12. Irreducible Maximal Finite Integral Matrix Groups
F 48.12. ImfNumberQQClasses
F 48.12. ImfNumberQClasses
F 48.12. ImfNumberZClasses
F 48.12. DisplayImfInvariants
F 48.12. DisplayImfInvariants
F 48.12. ImfInvariants
F 48.12. ImfInvariants
F 48.12. ImfMatrixGroup
F 48.12. ImfMatrixGroup
F 48.12. IsomorphismPermGroup!for Imf matrix groups
F 48.12. IsomorphismPermGroupImfGroup
C semigrp.tex 49. Semigroups
F 49.0. IsSemigroup
I 49.0. semigroup
F 49.0. Semigroup
F 49.0. Semigroup
F 49.0. Subsemigroup
F 49.0. SubsemigroupNC
F 49.0. SemigroupByGenerators
F 49.0. AsSemigroup
F 49.0. AsSubsemigroup
F 49.0. GeneratorsOfSemigroup
F 49.0. FreeSemigroup!with examples
F 49.0. FreeSemigroup!with examples
F 49.0. FreeSemigroup!with examples
F 49.0. FreeSemigroup!with examples
F 49.0. FreeSemigroup!with examples
F 49.0. SemigroupByMultiplicationTable
F 49.0. IsRegularSemigroup
F 49.0. IsRegularSemigroupElement
F 49.0. IsSimpleSemigroup
F 49.0. IsZeroSimpleSemigroup
F 49.0. IsZeroGroup
F 49.0. IsReesCongruenceSemigroup
S 49.1. Making transformation semigroups
F 49.1. IsTransformationSemigroup
F 49.1. IsTransformationMonoid
F 49.1. DegreeOfTransformationSemigroup
F 49.1. IsomorphismTransformationSemigroup
F 49.1. HomomorphismTransformationSemigroup
F 49.1. IsFullTransformationSemigroup
F 49.1. FullTransformationSemigroup
S 49.2. Ideals of semigroups
F 49.2. SemigroupIdealByGenerators
F 49.2. ReesCongruenceOfSemigroupIdeal
F 49.2. IsLeftSemigroupIdeal
F 49.2. IsRightSemigroupIdeal
F 49.2. IsSemigroupIdeal
S 49.3. Congruences for semigroups
F 49.3. IsSemigroupCongruence
F 49.3. IsReesCongruence
S 49.4. Quotients
F 49.4. IsQuotientSemigroup
F 49.4. HomomorphismQuotientSemigroup
F 49.4. QuotientSemigroupPreimage
F 49.4. QuotientSemigroupCongruence
F 49.4. QuotientSemigroupHomomorphism
S 49.5. Green's Relations
F 49.5. GreensRRelation
F 49.5. GreensLRelation
F 49.5. GreensJRelation
F 49.5. GreensDRelation
F 49.5. GreensHRelation
F 49.5. IsGreensRelation
F 49.5. IsGreensRRelation
F 49.5. IsGreensLRelation
F 49.5. IsGreensJRelation
F 49.5. IsGreensHRelation
F 49.5. IsGreensDRelation
F 49.5. IsGreensClass
F 49.5. IsGreensRClass
F 49.5. IsGreensLClass
F 49.5. IsGreensJClass
F 49.5. IsGreensHClass
F 49.5. IsGreensDClass
F 49.5. IsGreensLessThanOrEqual
F 49.5. RClassOfHClass
F 49.5. LClassOfHClass
F 49.5. EggBoxOfDClass
F 49.5. DisplayEggBoxOfDClass
F 49.5. GreensRClassOfElement
F 49.5. GreensLClassOfElement
F 49.5. GreensDClassOfElement
F 49.5. GreensJClassOfElement
F 49.5. GreensHClassOfElement
F 49.5. GreensRClasses
F 49.5. GreensLClasses
F 49.5. GreensJClasses
F 49.5. GreensDClasses
F 49.5. GreensHClasses
F 49.5. GroupHClassOfGreensDClass
F 49.5. IsGroupHClass
F 49.5. IsRegularDClass
S 49.6. Rees Matrix Semigroups
F 49.6. ReesMatrixSemigroup
F 49.6. ReesZeroMatrixSemigroup
F 49.6. IsReesMatrixSemigroup
F 49.6. IsReesZeroMatrixSemigroup
F 49.6. ReesMatrixSemigroupElement
F 49.6. ReesZeroMatrixSemigroupElement
F 49.6. IsReesMatrixSemigroupElement
F 49.6. IsReesZeroMatrixSemigroupElement
F 49.6. SandwichMatrixOfReesMatrixSemigroup
F 49.6. SandwichMatrixOfReesZeroMatrixSemigroup
F 49.6. RowIndexOfReesMatrixSemigroupElement
F 49.6. RowIndexOfReesZeroMatrixSemigroupElement
F 49.6. ColumnIndexOfReesMatrixSemigroupElement
F 49.6. ColumnIndexOfReesZeroMatrixSemigroupElement
F 49.6. UnderlyingElementOfReesMatrixSemigroupElement
F 49.6. UnderlyingElementOfReesZeroMatrixSemigroupElement
F 49.6. ReesZeroMatrixSemigroupElementIsZero
F 49.6. AssociatedReesMatrixSemigroupOfDClass
F 49.6. IsomorphismReesMatrixSemigroup
C monoid.tex 50. Monoids
F 50.0. IsMonoid
F 50.0. Monoid
F 50.0. Monoid
F 50.0. Monoid
F 50.0. Submonoid
F 50.0. SubmonoidNC
F 50.0. MonoidByGenerators
F 50.0. MonoidByGenerators
F 50.0. AsMonoid
F 50.0. AsSubmonoid
F 50.0. GeneratorsOfMonoid
F 50.0. TrivialSubmonoid
F 50.0. FreeMonoid
F 50.0. FreeMonoid
F 50.0. FreeMonoid
F 50.0. FreeMonoid
F 50.0. FreeMonoid
F 50.0. MonoidByMultiplicationTable
C fpsemi.tex 51. Finitely Presented Semigroups and Monoids
F 51.0. IsSubsemigroupFpSemigroup
F 51.0. IsSubmonoidFpMonoid
F 51.0. IsFpSemigroup
F 51.0. IsFpMonoid
F 51.0. IsElementOfFpSemigroup
F 51.0. IsElementOfFpMonoid
F 51.0. FpGrpMonSmgOfFpGrpMonSmgElement
S 51.1. Creating Finitely Presented Semigroups
F 51.1. quotient!of free semigroup
F 51.1. FactorFreeSemigroupByRelations
F 51.1. IsomorphismFpSemigroup
S 51.2. Comparison of Elements of Finitely Presented Semigroups
F 51.2. comparison!fp semigroup elements
S 51.3. Preimages in the Free Semigroup
F 51.3. FreeSemigroupOfFpSemigroup
F 51.3. FreeGeneratorsOfFpSemigroup
F 51.3. RelationsOfFpSemigroup
F 51.3. UnderlyingElement!fp semigroup elements
F 51.3. ElementOfFpSemigroup
S 51.4. Finitely presented monoids
F 51.4. quotient!of free monoid
S 51.5. Rewriting Systems and the Knuth-Bendix Procedure
F 51.5. ReducedConfluentRewritingSystem
F 51.5. ReducedConfluentRewritingSystem
F 51.5. KB_REW
F 51.5. GAPKB_REW
F 51.5. KnuthBendixRewritingSystem
F 51.5. KnuthBendixRewritingSystem
F 51.5. SemigroupOfRewritingSystem
F 51.5. MonoidOfRewritingSystem
F 51.5. FreeSemigroupOfRewritingSystem
F 51.5. FreeMonoidOfRewritingSystem
S 51.6. Todd-Coxeter Procedure
F 51.6. CosetTableOfFpSemigroup
C trans.tex 52. Transformations
F 52.0. IsTransformation
F 52.0. IsTransformationCollection
F 52.0. TransformationFamily
F 52.0. TransformationType
F 52.0. TransformationData
F 52.0. Transformation
F 52.0. TransformationNC
F 52.0. IdentityTransformation
F 52.0. RandomTransformation
F 52.0. DegreeOfTransformation
F 52.0. ImageListOfTransformation
F 52.0. ImageSetOfTransformation
F 52.0. RankOfTransformation
F 52.0. KernelOfTransformation
F 52.0. PreimagesOfTransformation
F 52.0. RestrictedTransformation
F 52.0. AsTransformation
F 52.0. AsTransformation
F 52.0. AsTransformationNC
F 52.0. PermLeftQuoTransformation
F 52.0. BinaryRelationTransformation
F 52.0. TransformationRelation
C addmagma.tex 53. Additive Magmas (preliminary)
S 53.1. (Near-)Additive Magma Categories
F 53.1. IsNearAdditiveMagma
F 53.1. IsNearAdditiveMagmaWithZero
F 53.1. IsNearAdditiveGroup
F 53.1. IsNearAdditiveMagmaWithInverses
F 53.1. IsAdditiveMagma
F 53.1. IsAdditiveMagmaWithZero
F 53.1. IsAdditiveGroup
F 53.1. IsAdditiveMagmaWithInverses
S 53.2. (Near-)Additive Magma Generation
F 53.2. NearAdditiveMagma
F 53.2. NearAdditiveMagma
F 53.2. NearAdditiveMagmaWithZero
F 53.2. NearAdditiveMagmaWithZero
F 53.2. NearAdditiveGroup
F 53.2. NearAdditiveGroup
F 53.2. NearAdditiveMagmaByGenerators
F 53.2. NearAdditiveMagmaByGenerators
F 53.2. NearAdditiveMagmaWithZeroByGenerators
F 53.2. NearAdditiveMagmaWithZeroByGenerators
F 53.2. NearAdditiveGroupByGenerators
F 53.2. NearAdditiveGroupByGenerators
F 53.2. SubnearAdditiveMagma
F 53.2. SubnearAdditiveMagmaNC
F 53.2. SubnearAdditiveMagmaWithZero
F 53.2. SubnearAdditiveMagmaWithZeroNC
F 53.2. SubnearAdditiveGroup
F 53.2. SubnearAdditiveGroupNC
S 53.3. Attributes and Properties for (Near-)Additive Magmas
F 53.3. IsAdditivelyCommutative
F 53.3. GeneratorsOfNearAdditiveMagma
F 53.3. GeneratorsOfAdditiveMagma
F 53.3. GeneratorsOfNearAdditiveMagmaWithZero
F 53.3. GeneratorsOfAdditiveMagmaWithZero
F 53.3. GeneratorsOfNearAdditiveGroup
F 53.3. GeneratorsOfAdditiveGroup
F 53.3. AdditiveNeutralElement
F 53.3. TrivialSubnearAdditiveMagmaWithZero
S 53.4. Operations for (Near-)Additive Magmas
F 53.4. ClosureNearAdditiveGroup
F 53.4. ClosureNearAdditiveGroup
C rings.tex 54. Rings
S 54.1. Generating Rings
F 54.1. IsRing
F 54.1. Ring
F 54.1. Ring
F 54.1. DefaultRing
F 54.1. DefaultRing
F 54.1. RingByGenerators
F 54.1. DefaultRingByGenerators
F 54.1. GeneratorsOfRing
F 54.1. AsRing
F 54.1. Subring
F 54.1. SubringNC
F 54.1. ClosureRing
F 54.1. ClosureRing
F 54.1. Quotient
F 54.1. Quotient
S 54.2. Ideals in Rings
F 54.2. TwoSidedIdeal
F 54.2. Ideal
F 54.2. LeftIdeal
F 54.2. RightIdeal
F 54.2. TwoSidedIdealNC
F 54.2. IdealNC
F 54.2. LeftIdealNC
F 54.2. RightIdealNC
F 54.2. IsTwoSidedIdeal
F 54.2. IsLeftIdeal
F 54.2. IsRightIdeal
F 54.2. IsTwoSidedIdealInParent
F 54.2. IsLeftIdealInParent
F 54.2. IsRightIdealInParent
F 54.2. TwoSidedIdealByGenerators
F 54.2. IdealByGenerators
F 54.2. LeftIdealByGenerators
F 54.2. RightIdealByGenerators
F 54.2. GeneratorsOfTwoSidedIdeal
F 54.2. GeneratorsOfIdeal
F 54.2. GeneratorsOfLeftIdeal
F 54.2. GeneratorsOfRightIdeal
F 54.2. LeftActingRingOfIdeal
F 54.2. RightActingRingOfIdeal
F 54.2. AsLeftIdeal
F 54.2. AsRightIdeal
F 54.2. AsTwoSidedIdeal
S 54.3. Rings With One
F 54.3. IsRingWithOne
F 54.3. RingWithOne
F 54.3. RingWithOne
F 54.3. RingWithOneByGenerators
F 54.3. GeneratorsOfRingWithOne
F 54.3. SubringWithOne
F 54.3. SubringWithOneNC
S 54.4. Properties of Rings
F 54.4. IsIntegralRing
F 54.4. IsUniqueFactorizationRing
F 54.4. IsLDistributive
F 54.4. IsRDistributive
F 54.4. IsDistributive
F 54.4. IsAnticommutative
F 54.4. IsZeroSquaredRing
F 54.4. IsJacobianRing
S 54.5. Units and Factorizations
F 54.5. IsUnit
F 54.5. IsUnit
F 54.5. Units
F 54.5. IsAssociated
F 54.5. IsAssociated
F 54.5. Associates
F 54.5. Associates
F 54.5. StandardAssociate
F 54.5. StandardAssociate
F 54.5. IsIrreducibleRingElement
F 54.5. IsIrreducibleRingElement
F 54.5. IsPrime
F 54.5. IsPrime
F 54.5. Factors
F 54.5. Factors
S 54.6. Euclidean Rings
F 54.6. IsEuclideanRing
F 54.6. EuclideanDegree
F 54.6. EuclideanDegree
F 54.6. EuclideanQuotient
F 54.6. EuclideanQuotient
F 54.6. EuclideanRemainder
F 54.6. EuclideanRemainder
F 54.6. QuotientRemainder
F 54.6. QuotientRemainder
S 54.7. Gcd and Lcm
F 54.7. Gcd
F 54.7. Gcd
F 54.7. Gcd
F 54.7. Gcd
F 54.7. GcdOp
F 54.7. GcdOp
F 54.7. GcdRepresentation
F 54.7. GcdRepresentation
F 54.7. GcdRepresentation
F 54.7. GcdRepresentation
F 54.7. GcdRepresentationOp
F 54.7. GcdRepresentationOp
F 54.7. Lcm
F 54.7. Lcm
F 54.7. Lcm
F 54.7. Lcm
F 54.7. LcmOp
F 54.7. LcmOp
F 54.7. QuotientMod
F 54.7. QuotientMod
F 54.7. PowerMod
F 54.7. PowerMod
F 54.7. InterpolatedPolynomial
C module.tex 55. Modules (preliminary)
S 55.1. Generating modules
F 55.1. IsLeftOperatorAdditiveGroup
F 55.1. IsLeftModule
F 55.1. GeneratorsOfLeftOperatorAdditiveGroup
F 55.1. GeneratorsOfLeftModule
F 55.1. AsLeftModule
F 55.1. IsRightOperatorAdditiveGroup
F 55.1. IsRightModule
F 55.1. GeneratorsOfRightOperatorAdditiveGroup
F 55.1. GeneratorsOfRightModule
F 55.1. LeftModuleByGenerators
F 55.1. LeftModuleByGenerators
F 55.1. LeftActingDomain
S 55.2. Submodules
F 55.2. Submodule
F 55.2. Submodule
F 55.2. SubmoduleNC
F 55.2. SubmoduleNC
F 55.2. ClosureLeftModule
F 55.2. TrivialSubmodule
S 55.3. Free Modules
F 55.3. IsFreeLeftModule
F 55.3. FreeLeftModule
F 55.3. FreeLeftModule
F 55.3. FreeLeftModule
F 55.3. FreeLeftModule
F 55.3. AsFreeLeftModule
F 55.3. Dimension
F 55.3. IsFiniteDimensional
F 55.3. UseBasis
F 55.3. IsRowModule
F 55.3. IsMatrixModule
F 55.3. IsFullRowModule
F 55.3. FullRowModule
F 55.3. IsFullMatrixModule
F 55.3. FullMatrixModule
F 55.3. IsHandledByNiceBasis
C fields.tex 56. Fields and Division Rings
I 56.0. fields
I 56.0. division rings
S 56.1. Generating Fields
F 56.1. IsDivisionRing
F 56.1. IsField
F 56.1. Field
F 56.1. Field
F 56.1. Field
F 56.1. DefaultField
F 56.1. DefaultField
F 56.1. DefaultFieldByGenerators
F 56.1. GeneratorsOfDivisionRing
F 56.1. GeneratorsOfField
F 56.1. DivisionRingByGenerators
F 56.1. DivisionRingByGenerators
F 56.1. AsDivisionRing
F 56.1. AsDivisionRing
F 56.1. AsField
F 56.1. AsField
S 56.2. Subfields of Fields
F 56.2. Subfield
F 56.2. SubfieldNC
F 56.2. FieldOverItselfByGenerators
F 56.2. PrimitiveElement
F 56.2. PrimeField
F 56.2. IsPrimeField
F 56.2. DegreeOverPrimeField
F 56.2. DefiningPolynomial
F 56.2. RootOfDefiningPolynomial
F 56.2. FieldExtension
F 56.2. Subfields
S 56.3. Galois Action
I 56.3. IsFieldControlledByGaloisGroup
F 56.3. GaloisGroup!of field
F 56.3. MinimalPolynomial!over a field
F 56.3. TracePolynomial
I 56.3. characteristic polynomial!for field elements
F 56.3. Norm
F 56.3. Norm
F 56.3. Norm
F 56.3. Trace!for field elements
F 56.3. Trace!for field elements
F 56.3. Trace!for field elements
F 56.3. Trace!for field elements
F 56.3. Conjugates
F 56.3. Conjugates
F 56.3. Conjugates
F 56.3. NormalBase
F 56.3. NormalBase
C fieldfin.tex 57. Finite Fields
S 57.1. Finite Field Elements
F 57.1. IsFFE
F 57.1. IsFFECollection
F 57.1. IsFFECollColl
F 57.1. Z
S 57.2. Operations for Finite Field Elements
F 57.2. DegreeFFE
F 57.2. DegreeFFE
F 57.2. DegreeFFE
F 57.2. LogFFE
F 57.2. IntFFE
F 57.2. IntFFESymm
F 57.2. IntFFESymm
F 57.2. IntVecFFE
S 57.3. Creating Finite Fields
I 57.3. DefaultField!for finite field elements
I 57.3. DefaultRing!for finite field elements
F 57.3. GaloisField
F 57.3. GF
F 57.3. GaloisField
F 57.3. GF
F 57.3. GaloisField
F 57.3. GF
F 57.3. GaloisField
F 57.3. GF
F 57.3. GaloisField
F 57.3. GF
F 57.3. PrimitiveRoot
S 57.4. FrobeniusAutomorphism
I 57.4. homomorphisms!Frobenius, field
I 57.4. field homomorphisms!Frobenius
I 57.4. Image!for Frobenius automorphisms
I 57.4. CompositionMapping!for Frobenius automorphisms
F 57.4. FrobeniusAutomorphism
I 57.4. Frobenius automorphism
S 57.5. Conway Polynomials
F 57.5. ConwayPolynomial
C fldabnum.tex 58. Abelian Number Fields
S 58.1. Construction of Abelian Number Fields
F 58.1. CyclotomicField
F 58.1. CyclotomicField
F 58.1. CyclotomicField
F 58.1. CyclotomicField
I 58.1. CF
F 58.1. AbelianNumberField
I 58.1. NF
F 58.1. GaussianRationals
F 58.1. IsGaussianRationals
S 58.2. Operations for Abelian Number Fields
I 58.2. cyclotomics!DefaultField
I 58.2. polynomials over abelian number fields!Factors
F 58.2. IsNumberField
I 58.2. number field
F 58.2. IsAbelianNumberField
I 58.2. abelian number field
F 58.2. IsCyclotomicField
F 58.2. GaloisStabilizer
S 58.3. Integral Bases of Abelian Number Fields
I 58.3. cyclotomic fields!CanonicalBasis
I 58.3. abelian number fields!CanonicalBasis
F 58.3. ZumbroichBase
F 58.3. LenstraBase
S 58.4. Galois Groups of Abelian Number Fields
I 58.4. abelian number fields!Galois group
I 58.4. number fields!Galois group
I 58.4. automorphism group!of number fields
F 58.4. ANFAutomorphism
S 58.5. Gaussians
F 58.5. GaussianIntegers
F 58.5. IsGaussianIntegers
C vspc.tex 59. Vector Spaces
F 59.0. IsLeftVectorSpace
F 59.0. IsVectorSpace
S 59.1. Constructing Vector Spaces
F 59.1. VectorSpace
F 59.1. Subspace
F 59.1. SubspaceNC
F 59.1. AsVectorSpace
F 59.1. AsSubspace
S 59.2. Operations and Attributes for Vector Spaces
F 59.2. GeneratorsOfLeftVectorSpace
F 59.2. GeneratorsOfVectorSpace
F 59.2. TrivialSubspace
S 59.3. Domains of Subspaces of Vector Spaces
F 59.3. Subspaces
F 59.3. Subspaces
F 59.3. IsSubspacesVectorSpace
S 59.4. Bases of Vector Spaces
F 59.4. IsBasis
F 59.4. Basis
F 59.4. Basis
F 59.4. BasisNC
F 59.4. CanonicalBasis
F 59.4. RelativeBasis
F 59.4. RelativeBasisNC
S 59.5. Operations for Vector Space Bases
F 59.5. BasisVectors
F 59.5. UnderlyingLeftModule
F 59.5. Coefficients
F 59.5. LinearCombination
F 59.5. LinearCombination
F 59.5. EnumeratorByBasis
F 59.5. IteratorByBasis
S 59.6. Operations for Special Kinds of Bases
F 59.6. IsCanonicalBasis
F 59.6. IsIntegralBasis
F 59.6. IsNormalBasis
F 59.6. StructureConstantsTable
S 59.7. Mutable Bases
F 59.7. IsMutableBasis
F 59.7. MutableBasis
F 59.7. NrBasisVectors
F 59.7. ImmutableBasis
F 59.7. IsContainedInSpan
F 59.7. CloseMutableBasis
S 59.8. Row and Matrix Spaces
I 59.8. row spaces
I 59.8. matrix spaces
F 59.8. IsRowSpace
F 59.8. IsMatrixSpace
F 59.8. IsGaussianSpace
F 59.8. FullRowSpace
F 59.8. FullMatrixSpace
F 59.8. DimensionOfVectors
F 59.8. IsSemiEchelonized
F 59.8. SemiEchelonBasis
F 59.8. SemiEchelonBasis
F 59.8. SemiEchelonBasisNC
I 59.8. canonical basis!for row spaces
F 59.8. IsCanonicalBasisFullRowModule
I 59.8. canonical basis!for matrix spaces
F 59.8. IsCanonicalBasisFullMatrixModule
F 59.8. NormedRowVectors
F 59.8. SiftedVector
S 59.9. Vector Space Homomorphisms
F 59.9. LeftModuleGeneralMappingByImages
F 59.9. LeftModuleHomomorphismByImages
F 59.9. LeftModuleHomomorphismByImagesNC
F 59.9. LeftModuleHomomorphismByMatrix
F 59.9. NaturalHomomorphismBySubspace
F 59.9. Hom
F 59.9. End
F 59.9. IsFullHomModule
F 59.9. IsPseudoCanonicalBasisFullHomModule
F 59.9. IsLinearMappingsModule
S 59.10. Vector Spaces Handled By Nice Bases
F 59.10. NiceFreeLeftModule
F 59.10. NiceVector
F 59.10. UglyVector
F 59.10. NiceFreeLeftModuleInfo
F 59.10. NiceBasis
F 59.10. IsBasisByNiceBasis
F 59.10. IsHandledByNiceBasis!for vector spaces
S 59.11. How to Implement New Kinds of Vector Spaces
F 59.11. DeclareHandlingByNiceBasis
F 59.11. InstallHandlingByNiceBasis
F 59.11. NiceBasisFiltersInfo
F 59.11. CheckForHandlingByNiceBasis
C algebra.tex 60. Algebras
F 60.0. InfoAlgebra
S 60.1. Constructing Algebras by Generators
F 60.1. Algebra
F 60.1. Algebra
F 60.1. Algebra
F 60.1. Algebra
F 60.1. AlgebraWithOne
F 60.1. AlgebraWithOne
F 60.1. AlgebraWithOne
F 60.1. AlgebraWithOne
S 60.2. Constructing Algebras as Free Algebras
F 60.2. FreeAlgebra
F 60.2. FreeAlgebra
F 60.2. FreeAlgebra
F 60.2. FreeAlgebraWithOne
F 60.2. FreeAlgebraWithOne
F 60.2. FreeAlgebraWithOne
F 60.2. FreeAssociativeAlgebra
F 60.2. FreeAssociativeAlgebra
F 60.2. FreeAssociativeAlgebra
F 60.2. FreeAssociativeAlgebraWithOne
F 60.2. FreeAssociativeAlgebraWithOne
F 60.2. FreeAssociativeAlgebraWithOne
S 60.3. Constructing Algebras by Structure Constants
F 60.3. EmptySCTable
F 60.3. EmptySCTable
F 60.3. EmptySCTable
F 60.3. SetEntrySCTable
F 60.3. GapInputSCTable
F 60.3. TestJacobi
F 60.3. AlgebraByStructureConstants
F 60.3. AlgebraByStructureConstants
F 60.3. AlgebraByStructureConstants
F 60.3. AlgebraByStructureConstants
F 60.3. IdentityFromSCTable
F 60.3. QuotientFromSCTable
S 60.4. Some Special Algebras
F 60.4. QuaternionAlgebra
F 60.4. QuaternionAlgebra
F 60.4. ComplexificationQuat
F 60.4. ComplexificationQuat
F 60.4. OctaveAlgebra
F 60.4. FullMatrixAlgebra
F 60.4. MatrixAlgebra
F 60.4. MatAlgebra
F 60.4. NullAlgebra
S 60.5. Subalgebras
F 60.5. Subalgebra
F 60.5. Subalgebra
F 60.5. SubalgebraNC
F 60.5. SubalgebraNC
F 60.5. SubalgebraWithOne
F 60.5. SubalgebraWithOne
F 60.5. SubalgebraWithOneNC
F 60.5. SubalgebraWithOneNC
F 60.5. TrivialSubalgebra
S 60.6. Ideals
S 60.7. Categories and Properties of Algebras
F 60.7. IsFLMLOR
F 60.7. IsFLMLORWithOne
F 60.7. IsAlgebra
F 60.7. IsAlgebraWithOne
F 60.7. IsLieAlgebra
F 60.7. IsSimpleAlgebra
F 60.7. IsFiniteDimensional!for matrix algebras
F 60.7. IsQuaternion
F 60.7. IsQuaternionCollection
F 60.7. IsQuaternionCollColl
S 60.8. Attributes and Operations for Algebras
F 60.8. GeneratorsOfAlgebra
F 60.8. GeneratorsOfAlgebraWithOne
F 60.8. ProductSpace
F 60.8. PowerSubalgebraSeries
F 60.8. AdjointBasis
F 60.8. IndicesOfAdjointBasis
F 60.8. AsAlgebra
F 60.8. AsAlgebraWithOne
F 60.8. AsSubalgebra
F 60.8. AsSubalgebraWithOne
F 60.8. MutableBasisOfClosureUnderAction
F 60.8. MutableBasisOfNonassociativeAlgebra
F 60.8. MutableBasisOfIdealInNonassociativeAlgebra
F 60.8. DirectSumOfAlgebras
F 60.8. DirectSumOfAlgebras
F 60.8. FullMatrixAlgebraCentralizer
F 60.8. RadicalOfAlgebra
F 60.8. CentralIdempotentsOfAlgebra
F 60.8. DirectSumDecomposition
F 60.8. LeviMalcevDecomposition
F 60.8. Grading
S 60.9. Homomorphisms of Algebras
F 60.9. AlgebraGeneralMappingByImages
F 60.9. AlgebraHomomorphismByImages
F 60.9. AlgebraHomomorphismByImagesNC
F 60.9. AlgebraWithOneGeneralMappingByImages
F 60.9. AlgebraWithOneHomomorphismByImages
F 60.9. AlgebraWithOneHomomorphismByImagesNC
F 60.9. NaturalHomomorphismByIdeal
F 60.9. OperationAlgebraHomomorphism
F 60.9. OperationAlgebraHomomorphism
F 60.9. IsomorphismFpAlgebra
F 60.9. IsomorphismMatrixAlgebra
F 60.9. IsomorphismSCAlgebra
F 60.9. IsomorphismSCAlgebra
F 60.9. RepresentativeLinearOperation
S 60.10. Representations of Algebras
F 60.10. LeftAlgebraModuleByGenerators
F 60.10. RightAlgebraModuleByGenerators
F 60.10. BiAlgebraModuleByGenerators
F 60.10. LeftAlgebraModule
F 60.10. RightAlgebraModule
F 60.10. BiAlgebraModule
F 60.10. GeneratorsOfAlgebraModule
F 60.10. IsAlgebraModuleElement
F 60.10. IsAlgebraModuleElementCollection
F 60.10. IsAlgebraModuleElementFamily
F 60.10. IsLeftAlgebraModuleElement
F 60.10. IsLeftAlgebraModuleElementCollection
F 60.10. IsRightAlgebraModuleElement
F 60.10. IsRightAlgebraModuleElementCollection
F 60.10. LeftActingAlgebra
F 60.10. RightActingAlgebra
F 60.10. ActingAlgebra
F 60.10. IsBasisOfAlgebraModuleElementSpace
F 60.10. MatrixOfAction
F 60.10. MatrixOfAction
F 60.10. SubAlgebraModule
F 60.10. LeftModuleByHomomorphismToMatAlg
F 60.10. RightModuleByHomomorphismToMatAlg
F 60.10. AdjointModule
F 60.10. FaithfulModule
F 60.10. ModuleByRestriction
F 60.10. ModuleByRestriction
F 60.10. NaturalHomomorphismBySubAlgebraModule
F 60.10. DirectSumOfAlgebraModules
F 60.10. DirectSumOfAlgebraModules
F 60.10. TranslatorSubalgebra
C alglie.tex 61. Lie Algebras
S 61.1. Lie objects
F 61.1. LieObject
F 61.1. IsLieObject
F 61.1. IsLieObjectCollection
F 61.1. LieFamily
I 61.1. Embedding!for Lie algebras
F 61.1. UnderlyingFamily
S 61.2. Constructing Lie algebras
F 61.2. LieAlgebraByStructureConstants
F 61.2. LieAlgebraByStructureConstants
F 61.2. LieAlgebraByStructureConstants
F 61.2. LieAlgebra
F 61.2. LieAlgebra
F 61.2. LieAlgebra
F 61.2. LieAlgebra
F 61.2. LieAlgebra
F 61.2. FreeLieAlgebra
F 61.2. FreeLieAlgebra
F 61.2. FreeLieAlgebra
F 61.2. FullMatrixLieAlgebra
F 61.2. MatrixLieAlgebra
F 61.2. MatLieAlgebra
F 61.2. RightDerivations
F 61.2. LeftDerivations
F 61.2. Derivations
F 61.2. SimpleLieAlgebra
S 61.3. Distinguished Subalgebras
F 61.3. LieCentre
F 61.3. LieCenter
F 61.3. LieCentralizer
F 61.3. LieNormalizer
F 61.3. LieDerivedSubalgebra
F 61.3. LieNilRadical
F 61.3. LieSolvableRadical
F 61.3. CartanSubalgebra
S 61.4. Series of Ideals
F 61.4. LieDerivedSeries
F 61.4. LieLowerCentralSeries
F 61.4. LieUpperCentralSeries
S 61.5. Properties of a Lie Algebra
F 61.5. IsLieAbelian
F 61.5. IsLieNilpotent
F 61.5. IsLieSolvable
S 61.6. Direct Sum Decompositions
F 61.6. LeviMalcevDecomposition!for Lie algebras
F 61.6. DirectSumDecomposition!for Lie algebras
S 61.7. Semisimple Lie Algebras and Root Systems
F 61.7. SemiSimpleType
F 61.7. ChevalleyBasis
F 61.7. IsRootSystem
F 61.7. IsRootSystemFromLieAlgebra
F 61.7. RootSystem
F 61.7. UnderlyingLieAlgebra
F 61.7. PositiveRoots
F 61.7. NegativeRoots
F 61.7. PositiveRootVectors
F 61.7. NegativeRootVectors
F 61.7. SimpleSystem
F 61.7. CartanMatrix
F 61.7. BilinearFormMat
F 61.7. CanonicalGenerators
F 61.7. IsWeylGroup
F 61.7. SparseCartanMatrix
F 61.7. WeylGroup
F 61.7. ApplySimpleReflection
F 61.7. LongestWeylWordPerm
F 61.7. ConjugateDominantWeight
F 61.7. ConjugateDominantWeightWithWord
F 61.7. WeylOrbitIterator
S 61.8. Restricted Lie algebras
F 61.8. IsRestrictedLieAlgebra
F 61.8. PthPowerImages
F 61.8. PthPowerImage
F 61.8. JenningsLieAlgebra
F 61.8. PCentralLieAlgebra
S 61.9. The Adjoint Representation
F 61.9. AdjointMatrix
F 61.9. AdjointAssociativeAlgebra
F 61.9. KillingMatrix
F 61.9. KappaPerp
F 61.9. IsNilpotentElement
F 61.9. NonNilpotentElement
F 61.9. FindSl2
S 61.10. Universal Enveloping Algebras
F 61.10. UniversalEnvelopingAlgebra
F 61.10. UniversalEnvelopingAlgebra
S 61.11. Finitely Presented Lie Algebras
F 61.11. FpLieAlgebraByCartanMatrix
F 61.11. NilpotentQuotientOfFpLieAlgebra
F 61.11. NilpotentQuotientOfFpLieAlgebra
S 61.12. Modules over Lie Algebras and Their Cohomology
F 61.12. FaithfulModule!for Lie algebras
F 61.12. IsCochain
F 61.12. IsCochainCollection
F 61.12. Cochain
F 61.12. CochainSpace
F 61.12. ValueCochain
F 61.12. LieCoboundaryOperator
F 61.12. Cocycles
F 61.12. Coboundaries
S 61.13. Modules over Semisimple Lie Algebras
F 61.13. DominantWeights
F 61.13. DominantCharacter
F 61.13. DominantCharacter
F 61.13. DecomposeTensorProduct
F 61.13. DimensionOfHighestWeightModule
F 61.13. IsUEALatticeElement
F 61.13. IsUEALatticeElementCollection
F 61.13. IsUEALatticeElementFamily
F 61.13. LatticeGeneratorsInUEA
F 61.13. ObjByExtRep
F 61.13. IsWeightRepElement
F 61.13. IsWeightRepElementCollection
F 61.13. IsWeightRepElementFamily
F 61.13. HighestWeightModule
S 61.14. Tensor Products and Exterior and Symmetric Powers
F 61.14. TensorProductOfAlgebraModules
F 61.14. TensorProductOfAlgebraModules
F 61.14. ExteriorPowerOfAlgebraModule
F 61.14. SymmetricPowerOfAlgebraModule
F 61.14. DirectSumOfAlgebraModules!for Lie algebras
F 61.14. DirectSumOfAlgebraModules!for Lie algebras
C algfp.tex 62. Finitely Presented Algebras
C mgmring.tex 63. Magma Rings
I 63.0. group algebra
I 63.0. group ring
S 63.1. Free Magma Rings
F 63.1. FreeMagmaRing
F 63.1. GroupRing
F 63.1. IsFreeMagmaRing
F 63.1. IsFreeMagmaRingWithOne
F 63.1. IsGroupRing
F 63.1. UnderlyingMagma
F 63.1. AugmentationIdeal
S 63.2. Elements of Free Magma Rings
F 63.2. IsElementOfFreeMagmaRing
F 63.2. IsElementOfFreeMagmaRingCollection
F 63.2. IsElementOfFreeMagmaRingFamily
F 63.2. CoefficientsAndMagmaElements
F 63.2. ZeroCoefficient
F 63.2. ElementOfMagmaRing
S 63.3. Natural Embeddings related to Magma Rings
I 63.3. Embedding!for magma rings
S 63.4. Magma Rings modulo Relations
F 63.4. IsElementOfMagmaRingModuloRelations
F 63.4. IsElementOfMagmaRingModuloRelationsCollection
F 63.4. IsElementOfMagmaRingModuloRelationsFamily
F 63.4. NormalizedElementOfMagmaRingModuloRelations
F 63.4. IsMagmaRingModuloRelations
S 63.5. Magma Rings modulo the Span of a Zero Element
F 63.5. IsElementOfMagmaRingModuloSpanOfZeroFamily
F 63.5. IsMagmaRingModuloSpanOfZero
F 63.5. MagmaRingModuloSpanOfZero
S 63.6. Technical Details about the Implementation of Magma Rings
C ratfun.tex 64. Polynomials and Rational Functions
S 64.1. Indeterminates
F 64.1. Indeterminate
F 64.1. Indeterminate
F 64.1. Indeterminate
F 64.1. Indeterminate
F 64.1. IndeterminateNumberOfUnivariateRationalFunction
F 64.1. IndeterminateOfUnivariateRationalFunction
F 64.1. IndeterminateName
F 64.1. HasIndeterminateName
F 64.1. SetIndeterminateName
F 64.1. CIUnivPols
S 64.2. Operations for Rational Functions
F 64.2. addition!rational functions
F 64.2. subtraction!rational functions
F 64.2. product!rational functions
F 64.2. quotient!rational functions
F 64.2. mod!Laurent polynomials
S 64.3. Comparison of Rational Functions
F 64.3. comparison!rational functions
F 64.3. smaller!rational functions
S 64.4. Properties and Attributes of Rational Functions
F 64.4. IsPolynomialFunction
F 64.4. IsRationalFunction
F 64.4. NumeratorOfRationalFunction
F 64.4. DenominatorOfRationalFunction
F 64.4. IsPolynomial
F 64.4. AsPolynomial
F 64.4. IsUnivariateRationalFunction
F 64.4. CoefficientsOfUnivariateRationalFunction
F 64.4. IsUnivariatePolynomial
F 64.4. CoefficientsOfUnivariatePolynomial
F 64.4. IsLaurentPolynomial
F 64.4. IsConstantRationalFunction
F 64.4. IsPrimitivePolynomial
F 64.4. SplittingField
S 64.5. Univariate Polynomials
F 64.5. UnivariatePolynomial
F 64.5. UnivariatePolynomialByCoefficients
F 64.5. DegreeOfLaurentPolynomial
F 64.5. RootsOfUPol
F 64.5. RootsOfUPol
F 64.5. RootsOfUPol
F 64.5. UnivariatenessTestRationalFunction
S 64.6. Polynomials as Univariate Polynomials in one Indeterminate
F 64.6. DegreeIndeterminate
F 64.6. DegreeIndeterminate
F 64.6. PolynomialCoefficientsOfPolynomial
F 64.6. PolynomialCoefficientsOfPolynomial
F 64.6. LeadingCoefficient
F 64.6. LeadingMonomial
F 64.6. Derivative
F 64.6. Derivative
F 64.6. Discriminant
F 64.6. Discriminant
F 64.6. Resultant
F 64.6. Resultant
S 64.7. Multivariate Polynomials
F 64.7. Value
F 64.7. Value
F 64.7. OnIndeterminates
S 64.8. Minimal Polynomials
I 64.8. MinimalPolynomial!over a ring
F 64.8. MinimalPolynomial
S 64.9. Cyclotomic Polynomials
F 64.9. CyclotomicPolynomial
S 64.10. Polynomial Factorization
F 64.10. Factors!of univariate polynomial
F 64.10. FactorsSquarefree
S 64.11. Polynomials over the Rationals
F 64.11. PrimitivePolynomial
F 64.11. PolynomialModP
F 64.11. GaloisType
F 64.11. ProbabilityShapes
F 64.11. BombieriNorm
F 64.11. MinimizedBombieriNorm
F 64.11. HenselBound
F 64.11. OneFactorBound
S 64.12. Laurent Polynomials
F 64.12. LaurentPolynomialByCoefficients
F 64.12. CoefficientsOfLaurentPolynomial
F 64.12. IndeterminateNumberOfLaurentPolynomial
F 64.12. QuotRemLaurpols
S 64.13. Univariate Rational Functions
F 64.13. UnivariateRationalFunctionByCoefficients
S 64.14. Polynomial Rings
F 64.14. PolynomialRing
F 64.14. PolynomialRing
F 64.14. PolynomialRing
F 64.14. PolynomialRing
F 64.14. IndeterminatesOfPolynomialRing
F 64.14. CoefficientsRing
F 64.14. IsPolynomialRing
F 64.14. IsFiniteFieldPolynomialRing
F 64.14. IsAbelianNumberFieldPolynomialRing
F 64.14. IsRationalsPolynomialRing
S 64.15. Univariate Polynomial Rings
F 64.15. UnivariatePolynomialRing
F 64.15. UnivariatePolynomialRing
F 64.15. UnivariatePolynomialRing
F 64.15. IsUnivariatePolynomialRing
S 64.16. Monomial Orderings
F 64.16. IsMonomialOrdering
F 64.16. LeadingMonomialOfPolynomial
F 64.16. LeadingTermOfPolynomial
F 64.16. LeadingCoefficientOfPolynomial
F 64.16. MonomialComparisonFunction
F 64.16. MonomialExtrepComparisonFun
F 64.16. MonomialLexOrdering
F 64.16. MonomialLexOrdering
F 64.16. MonomialGrlexOrdering
F 64.16. MonomialGrlexOrdering
F 64.16. MonomialGrevlexOrdering
F 64.16. MonomialGrevlexOrdering
F 64.16. EliminationOrdering
F 64.16. EliminationOrdering
F 64.16. PolynomialReduction
F 64.16. PolynomialReducedRemainder
F 64.16. PolynomialDivisionAlgorithm
F 64.16. MonomialExtGrlexLess
S 64.17. Groebner Bases
F 64.17. GroebnerBasis
F 64.17. GroebnerBasis
F 64.17. GroebnerBasisNC
F 64.17. ReducedGroebnerBasis
F 64.17. ReducedGroebnerBasis
F 64.17. StoredGroebnerBasis
F 64.17. InfoGroebner
S 64.18. Rational Function Families
F 64.18. RationalFunctionsFamily
F 64.18. IsPolynomialFunctionsFamily
F 64.18. IsRationalFunctionsFamily
F 64.18. CoefficientsFamily
S 64.19. The Representations of Rational Functions
S 64.20. The Defining Attributes of Rational Functions
I 64.20. Expanded form of monomials
I 64.20. External representation of polynomials
F 64.20. IsRationalFunctionDefaultRep
F 64.20. ExtRepNumeratorRatFun
F 64.20. ExtRepDenominatorRatFun
F 64.20. ZeroCoefficientRatFun
F 64.20. IsPolynomialDefaultRep
F 64.20. ExtRepPolynomialRatFun
F 64.20. IsLaurentPolynomialDefaultRep
S 64.21. Creation of Rational Functions
F 64.21. RationalFunctionByExtRep
F 64.21. PolynomialByExtRep
F 64.21. LaurentPolynomialByExtRep
S 64.22. Arithmetic for External Representations of Polynomials
F 64.22. ZippedSum
F 64.22. ZippedProduct
F 64.22. QuotientPolynomialsExtRep
S 64.23. Cancellation Tests for Rational Functions
F 64.23. RationalFunctionByExtRepWithCancellation
F 64.23. TryGcdCancelExtRepPolynomials
F 64.23. HeuristicCancelPolynomials
C algfld.tex 65. Algebraic extensions of fields
S 65.1. Creation of Algebraic Extensions
F 65.1. AlgebraicExtension
F 65.1. IsAlgebraicExtension
S 65.2. Elements in Algebraic Extensions
I 65.2. Operations for algebraic elements
F 65.2. IsAlgebraicElement
C padics.tex 66. p-adic Numbers (preliminary)
S 66.1. Pure p-adic Numbers
F 66.1. PurePadicNumberFamily
F 66.1. PadicNumber!for pure padics
F 66.1. Valuation
F 66.1. ShiftedPadicNumber
F 66.1. IsPurePadicNumber
F 66.1. IsPurePadicNumberFamily
S 66.2. Extensions of the p-adic Numbers
F 66.2. PadicExtensionNumberFamily
F 66.2. PadicNumber
F 66.2. PadicNumber
F 66.2. PadicNumber
F 66.2. IsPadicExtensionNumber
F 66.2. IsPadicExtensionNumberFamily
C meataxe.tex 67. The MeatAxe
S 67.1. MeatAxe Modules
F 67.1. GModuleByMats
F 67.1. GModuleByMats
S 67.2. Module Constructions
F 67.2. TensorProductGModule 
F 67.2. WedgeGModule 
S 67.3. Selecting a Different MeatAxe
S 67.4. Accessing a Module
F 67.4. MTX.Generators
F 67.4. MTX.Dimension
F 67.4. MTX.Field
S 67.5. Irreducibility Tests
F 67.5. MTX.IsIrreducible
F 67.5. MTX.IsAbsolutelyIrreducible
F 67.5. MTX.DegreeSplittingField
S 67.6. Finding Submodules
F 67.6. MTX.SubmoduleGModule
F 67.6. MTX.SubGModule
F 67.6. MTX.ProperSubmoduleBasis
F 67.6. MTX.BasesSubmodules
F 67.6. MTX.BasesMinimalSubmodules
F 67.6. MTX.BasesMaximalSubmodules
F 67.6. MTX.BasisRadical
F 67.6. MTX.BasisSocle
F 67.6. MTX.BasesMinimalSupermodules
F 67.6. MTX.BasesCompositionSeries
F 67.6. MTX.CompositionFactors
F 67.6. MTX.CollectedFactors
S 67.7. Induced Actions
F 67.7. MTX.NormedBasisAndBaseChange
F 67.7. MTX.InducedActionSubmodule
F 67.7. MTX.InducedActionSubmoduleNB
F 67.7. MTX.InducedActionFactorModule
F 67.7. MTX.InducedActionMatrix
F 67.7. MTX.InducedActionMatrixNB
F 67.7. MTX.InducedActionFactorMatrix
F 67.7. MTX.InducedAction
S 67.8. Module Homomorphisms
F 67.8. MTX.IsEquivalent
F 67.8. MTX.Isomorphism
F 67.8. MTX.Homomorphism
F 67.8. MTX.Homomorphisms
F 67.8. MTX.Distinguish
S 67.9. Invariant Forms
F 67.9. MTX.InvariantBilinearForm
F 67.9. MTX.InvariantSesquilinearForm
F 67.9. MTX.InvariantQuadraticForm
F 67.9. MTX.BasisInOrbit
F 67.9. MTX.OrthogonalSign
S 67.10. The Smash MeatAxe
F 67.10. SMTX.RandomIrreducibleSubGModule
F 67.10. SMTX.GoodElementGModule
F 67.10. SMTX.SortHomGModule
F 67.10. SMTX.MinimalSubGModules
F 67.10. SMTX.Setter
F 67.10. SMTX.Getter
F 67.10. SMTX.IrreducibilityTest
F 67.10. SMTX.AbsoluteIrreducibilityTest
F 67.10. SMTX.MinimalSubGModule
F 67.10. SMTX.MatrixSum
F 67.10. SMTX.CompleteBasis
S 67.11. Smash MeatAxe Flags
F 67.11. SMTX.Subbasis
F 67.11. SMTX.AlgEl
F 67.11. SMTX.AlgElMat
F 67.11. SMTX.AlgElCharPol
F 67.11. SMTX.AlgElCharPolFac
F 67.11. SMTX.AlgElNullspaceVec
F 67.11. SMTX.AlgElNullspaceDimension
F 67.11. SMTX.CentMat
F 67.11. SMTX.CentMatMinPoly
C tom.tex 68. Tables of Marks
S 68.1. More about Tables of Marks
S 68.2. Table of Marks Objects in GAP
S 68.3. Constructing Tables of Marks
F 68.3. TableOfMarks
F 68.3. TableOfMarks
F 68.3. TableOfMarks
F 68.3. TableOfMarksByLattice
F 68.3. LatticeSubgroupsByTom
S 68.4. Printing Tables of Marks
I 68.4. ViewObj!for tables of marks
I 68.4. PrintObj!for tables of marks
I 68.4. Display!for tables of marks
S 68.5. Sorting Tables of Marks
F 68.5. SortedTom
F 68.5. PermutationTom
S 68.6. Technical Details about Tables of Marks
F 68.6. InfoTom
F 68.6. IsTableOfMarks
F 68.6. TableOfMarksFamily
F 68.6. TableOfMarksComponents
F 68.6. ConvertToTableOfMarks
S 68.7. Attributes of Tables of Marks
F 68.7. MarksTom
F 68.7. SubsTom
F 68.7. NrSubsTom
F 68.7. OrdersTom
F 68.7. LengthsTom
F 68.7. ClassTypesTom
F 68.7. ClassNamesTom
F 68.7. FusionsTom
F 68.7. UnderlyingGroup!for tables of marks
F 68.7. IdempotentsTom
F 68.7. IdempotentsTomInfo
F 68.7. Identifier!for tables of marks
F 68.7. MatTom
F 68.7. MoebiusTom
F 68.7. WeightsTom
S 68.8. Properties of Tables of Marks
F 68.8. IsAbelianTom
F 68.8. IsCyclicTom
F 68.8. IsNilpotentTom
F 68.8. IsPerfectTom
F 68.8. IsSolvableTom
S 68.9. Other Operations for Tables of Marks
F 68.9. IsInternallyConsistent!for tables of marks
F 68.9. DerivedSubgroupTom
F 68.9. DerivedSubgroupsTom
F 68.9. DerivedSubgroupsTomPossible
F 68.9. DerivedSubgroupsTomUnique
F 68.9. NormalizerTom
F 68.9. NormalizersTom
F 68.9. ContainedTom
F 68.9. ContainingTom
F 68.9. CyclicExtensionsTom
F 68.9. CyclicExtensionsTom
F 68.9. CyclicExtensionsTom
F 68.9. DecomposedFixedPointVector
F 68.9. EulerianFunctionByTom
F 68.9. IntersectionsTom
F 68.9. FactorGroupTom
F 68.9. MaximalSubgroupsTom
F 68.9. MaximalSubgroupsTom
F 68.9. MinimalSupergroupsTom
S 68.10. Standard Generators of Groups
F 68.10. StandardGeneratorsInfo!for groups
F 68.10. HumanReadableDefinition
F 68.10. ScriptFromString
F 68.10. StandardGeneratorsFunctions
F 68.10. IsStandardGeneratorsOfGroup
F 68.10. StandardGeneratorsOfGroup
S 68.11. Accessing Subgroups via Tables of Marks
F 68.11. GeneratorsSubgroupsTom
F 68.11. StraightLineProgramsTom
F 68.11. IsTableOfMarksWithGens
F 68.11. RepresentativeTom
F 68.11. RepresentativeTomByGenerators
F 68.11. RepresentativeTomByGeneratorsNC
F 68.11. StandardGeneratorsInfo!for tables of marks
S 68.12. The Interface between Tables of Marks and Character Tables
F 68.12. FusionCharTableTom
F 68.12. PossibleFusionsCharTableTom
F 68.12. PermCharsTom
F 68.12. PermCharsTom
S 68.13. Generic Construction of Tables of Marks
F 68.13. TableOfMarksCyclic
F 68.13. TableOfMarksDihedral
F 68.13. TableOfMarksFrobenius
S 68.14. The Library of Tables of Marks
C ctbl.tex 69. Character Tables
I 69.0. tables
S 69.1. Some Remarks about Character Theory in GAP
S 69.2. History of Character Theory Stuff in GAP
S 69.3. Creating Character Tables
I 69.3. tables
I 69.3. character tables
I 69.3. library tables
I 69.3. character tables!access to
I 69.3. character tables!calculate
I 69.3. character tables!of groups
F 69.3. CharacterTable
F 69.3. CharacterTable
F 69.3. CharacterTable
F 69.3. CharacterTable
F 69.3. BrauerTable
F 69.3. BrauerTable
F 69.3. BrauerTableOp
F 69.3. ComputedBrauerTables
F 69.3. SupportedCharacterTableInfo
F 69.3. ConvertToCharacterTable
F 69.3. ConvertToCharacterTableNC
S 69.4. Character Table Categories
F 69.4. IsNearlyCharacterTable
F 69.4. IsCharacterTable
F 69.4. IsOrdinaryTable
F 69.4. IsBrauerTable
F 69.4. IsCharacterTableInProgress
F 69.4. InfoCharacterTable
F 69.4. NearlyCharacterTablesFamily
S 69.5. Conventions for Character Tables
S 69.6. The Interface between Character Tables and Groups
F 69.6. UnderlyingGroup!for character tables
F 69.6. ConjugacyClasses!for character tables
F 69.6. IdentificationOfConjugacyClasses
F 69.6. ConnectGroupAndCharacterTable
F 69.6. ConnectGroupAndCharacterTable
F 69.6. CompatibleConjugacyClasses
F 69.6. CompatibleConjugacyClasses
S 69.7. Operators for Character Tables
I 69.7. \*!for character tables
I 69.7. /!for character tables
I 69.7. mod!for character tables
I 69.7. character tables!infix operators
S 69.8. Attributes and Properties of Character Tables
F 69.8. CharacterDegrees
F 69.8. CharacterDegrees
F 69.8. CharacterDegrees
F 69.8. Irr
F 69.8. Irr
F 69.8. Irr
F 69.8. LinearCharacters
F 69.8. LinearCharacters
F 69.8. LinearCharacters
F 69.8. OrdinaryCharacterTable
F 69.8. OrdinaryCharacterTable
I 69.8. AbelianInvariants!for character tables
I 69.8. CommutatorLength!for character tables
I 69.8. Exponent!for character tables
I 69.8. IsAbelian!for character tables
I 69.8. IsCyclic!for character tables
I 69.8. IsFinite!for character tables
I 69.8. IsMonomial!for character tables
I 69.8. IsNilpotent!for character tables
I 69.8. IsPerfect!for character tables
I 69.8. IsSimple!for character tables
I 69.8. IsSolvable!for character tables
I 69.8. IsSporadicSimple!for character tables
I 69.8. IsSupersolvable!for character tables
I 69.8. NrConjugacyClasses!for character tables
I 69.8. Size!for character tables
F 69.8. OrdersClassRepresentatives
F 69.8. SizesCentralizers
F 69.8. SizesConjugacyClasses
F 69.8. AutomorphismsOfTable
F 69.8. UnderlyingCharacteristic
F 69.8. UnderlyingCharacteristic
F 69.8. ClassNames
F 69.8. ClassNames
F 69.8. CharacterNames
F 69.8. Identifier!for character tables
F 69.8. InfoText
F 69.8. InverseClasses
F 69.8. RealClasses
I 69.8. classes!real
F 69.8. ClassOrbit
F 69.8. ClassRoots
F 69.8. ClassPositionsOfNormalSubgroups
F 69.8. ClassPositionsOfMaximalNormalSubgroups
F 69.8. ClassPositionsOfAgemo
F 69.8. ClassPositionsOfCentre!for character tables
F 69.8. ClassPositionsOfDirectProductDecompositions
F 69.8. ClassPositionsOfDirectProductDecompositions
F 69.8. ClassPositionsOfDerivedSubgroup
F 69.8. ClassPositionsOfElementaryAbelianSeries
F 69.8. ClassPositionsOfFittingSubgroup
F 69.8. ClassPositionsOfLowerCentralSeries
F 69.8. ClassPositionsOfUpperCentralSeries
F 69.8. ClassPositionsOfSupersolvableResiduum
F 69.8. ClassPositionsOfNormalClosure
S 69.9. Operations Concerning Blocks
F 69.9. PrimeBlocks
F 69.9. PrimeBlocksOp
F 69.9. ComputedPrimeBlockss
F 69.9. SameBlock
F 69.9. BlocksInfo
F 69.9. DecompositionMatrix
F 69.9. DecompositionMatrix
F 69.9. LaTeXStringDecompositionMatrix
S 69.10. Other Operations for Character Tables
F 69.10. IsInternallyConsistent!for character tables
F 69.10. IsPSolvableCharacterTable
F 69.10. IsPSolvableCharacterTableOp
F 69.10. ComputedIsPSolvableCharacterTables
F 69.10. IsClassFusionOfNormalSubgroup
F 69.10. Indicator
F 69.10. Indicator
F 69.10. Indicator
F 69.10. IndicatorOp
F 69.10. ComputedIndicators
F 69.10. NrPolyhedralSubgroups
I 69.10. subgroups!polyhedral
F 69.10. ClassMultiplicationCoefficient!for character tables
I 69.10. class multiplication coefficient
I 69.10. structure constant
F 69.10. ClassStructureCharTable
I 69.10. class multiplication coefficient
I 69.10. structure constant
F 69.10. MatClassMultCoeffsCharTable
I 69.10. structure constant
I 69.10. class multiplication coefficient
S 69.11. Printing Character Tables
I 69.11. ViewObj!for character tables
I 69.11. PrintObj!for character tables
I 69.11. Display!for character tables
F 69.11. DisplayOptions
F 69.11. PrintCharacterTable
S 69.12. Computing the Irreducible Characters of a Group
F 69.12. IrrDixonSchneider
F 69.12. IrrConlon
F 69.12. IrrBaumClausen
F 69.12. IrreducibleRepresentations
F 69.12. IrreducibleRepresentations
F 69.12. IrreducibleRepresentationsDixon
F 69.12. IrreducibleRepresentationsDixon
F 69.12. IrreducibleRepresentationsDixon
S 69.13. Representations given by modules
F 69.13. IrreducibleModules
F 69.13. AbsoluteIrreducibleModules
F 69.13. AbsolutIrreducibleModules
F 69.13. RegularModule
S 69.14. The Dixon-Schneider Algorithm
I 69.14. Dixon-Schneider algorithm
S 69.15. Advanced Methods for Dixon-Schneider Calculations
I 69.15. irreducible characters!computation
F 69.15. DixonRecord
F 69.15. DixonInit
F 69.15. DixontinI
F 69.15. DixonSplit
F 69.15. BestSplittingMatrix
F 69.15. DxIncludeIrreducibles
F 69.15. SplitCharacters
F 69.15. IsDxLargeGroup
S 69.16. Components of a Dixon Record
S 69.17. An Example of Advanced Dixon-Schneider Calculations
S 69.18. Constructing Character Tables from Others
F 69.18. CharacterTableDirectProduct
F 69.18. FactorsOfDirectProduct
F 69.18. CharacterTableFactorGroup
F 69.18. CharacterTableIsoclinic
F 69.18. CharacterTableIsoclinic
F 69.18. CharacterTableIsoclinic
F 69.18. CharacterTableWreathSymmetric
S 69.19. Sorted Character Tables
F 69.19. CharacterTableWithSortedCharacters
F 69.19. CharacterTableWithSortedCharacters
F 69.19. SortedCharacters
F 69.19. SortedCharacters
F 69.19. SortedCharacters
F 69.19. CharacterTableWithSortedClasses
F 69.19. CharacterTableWithSortedClasses
F 69.19. CharacterTableWithSortedClasses
F 69.19. CharacterTableWithSortedClasses
F 69.19. SortedCharacterTable
F 69.19. SortedCharacterTable
F 69.19. SortedCharacterTable
F 69.19. ClassPermutation
S 69.20. Automorphisms and Equivalence of Character Tables
F 69.20. MatrixAutomorphisms
F 69.20. TableAutomorphisms
F 69.20. TableAutomorphisms
F 69.20. TableAutomorphisms
F 69.20. TransformingPermutations
F 69.20. TransformingPermutationsCharacterTables
F 69.20. FamiliesOfRows
S 69.21. Storing Normal Subgroup Information
F 69.21. NormalSubgroupClassesInfo
F 69.21. ClassPositionsOfNormalSubgroup
F 69.21. NormalSubgroupClasses
F 69.21. FactorGroupNormalSubgroupClasses
C ctblfuns.tex 70. Class Functions
I 70.0. characters
I 70.0. group characters
I 70.0. virtual characters
I 70.0. generalized characters
F 70.0. IsClassFunction
I 70.0. class function
I 70.0. class function objects
S 70.1. Why Class Functions?
S 70.2. Basic Operations for Class Functions
F 70.2. UnderlyingCharacterTable
F 70.2. ValuesOfClassFunction
S 70.3. Comparison of Class Functions
S 70.4. Arithmetic Operations for Class Functions
I 70.4. class functions!as ring elements
I 70.4. inverse!of class function
I 70.4. character value!of group element using powering operator
I 70.4. power!meaning for class functions
I 70.4. {\accent 94 }!for class functions
I 70.4. characteristic!for class functions
I 70.4. ComplexConjugate!for class functions
I 70.4. GaloisCyc!for class functions
I 70.4. Permuted!for class functions
I 70.4. Order!of a class function
S 70.5. Printing Class Functions
I 70.5. ViewObj!for class functions
I 70.5. PrintObj!for character tables
I 70.5. Display!for character tables
S 70.6. Creating Class Functions from Values Lists
F 70.6. ClassFunction
F 70.6. ClassFunction
F 70.6. VirtualCharacter
F 70.6. VirtualCharacter
F 70.6. Character
F 70.6. ClassFunctionSameType
S 70.7. Creating Class Functions using Groups
F 70.7. TrivialCharacter
F 70.7. TrivialCharacter
F 70.7. NaturalCharacter
F 70.7. NaturalCharacter
F 70.7. PermutationCharacter
F 70.7. PermutationCharacter
S 70.8. Operations for Class Functions
F 70.8. IsCharacter
I 70.8. ordinary character
I 70.8. Brauer character
F 70.8. IsVirtualCharacter
I 70.8. virtual character
F 70.8. IsIrreducibleCharacter
I 70.8. irreducible character
F 70.8. DegreeOfCharacter
I 70.8. constituent!of a group character
I 70.8. decompose!a group character
I 70.8. multiplicity!of constituents of a group character
I 70.8. inner product!of group characters
F 70.8. ScalarProduct!for characters
F 70.8. MatScalarProducts
F 70.8. MatScalarProducts
F 70.8. Norm!of character
F 70.8. ConstituentsOfCharacter
F 70.8. KernelOfCharacter
F 70.8. ClassPositionsOfKernel
F 70.8. CentreOfCharacter
I 70.8. centre!of a character
F 70.8. ClassPositionsOfCentre!for characters
F 70.8. InertiaSubgroup
F 70.8. CycleStructureClass
F 70.8. IsTransitive!for characters
I 70.8. IsTransitive!for class functions
F 70.8. Transitivity!for characters
I 70.8. Transitivity!for class functions
F 70.8. CentralCharacter
I 70.8. central character
F 70.8. DeterminantOfCharacter
I 70.8. determinant character
F 70.8. EigenvaluesChar
F 70.8. Tensored
S 70.9. Restricted and Induced Class Functions
I 70.9. inflated class functions
F 70.9. RestrictedClassFunction
F 70.9. RestrictedClassFunction
F 70.9. RestrictedClassFunction
F 70.9. RestrictedClassFunctions
F 70.9. RestrictedClassFunctions
F 70.9. RestrictedClassFunctions
F 70.9. InducedClassFunction
F 70.9. InducedClassFunction
F 70.9. InducedClassFunction
F 70.9. InducedClassFunctions
F 70.9. InducedClassFunctions
F 70.9. InducedClassFunctions
F 70.9. InducedCyclic
F 70.9. InducedCyclic
F 70.9. InducedCyclic
F 70.9. InducedCyclic
S 70.10. Reducing Virtual Characters
F 70.10. ReducedClassFunctions
F 70.10. ReducedClassFunctions
F 70.10. ReducedCharacters
F 70.10. IrreducibleDifferences
F 70.10. IrreducibleDifferences
F 70.10. IrreducibleDifferences
F 70.10. IrreducibleDifferences
F 70.10. LLL
I 70.10. LLL algorithm!for virtual characters
I 70.10. short vectors spanning a lattice
I 70.10. lattice basis reduction!for virtual characters
F 70.10. Extract
F 70.10. OrthogonalEmbeddingsSpecialDimension
F 70.10. Decreased
F 70.10. DnLattice
F 70.10. DnLatticeIterative
S 70.11. Symmetrizations of Class Functions
I 70.11. characters!symmetrizations of
F 70.11. Symmetrizations
F 70.11. Symmetrizations
F 70.11. SymmetricParts
F 70.11. AntiSymmetricParts
F 70.11. OrthogonalComponents
I 70.11. symmetrizations!orthogonal
I 70.11. Frame
I 70.11. Murnaghan components
I 70.11. symmetrizations!orthogonal
I 70.11. Frame
I 70.11. Murnaghan components
F 70.11. SymplecticComponents
I 70.11. symmetrizations!symplectic
I 70.11. Murnaghan components
S 70.12. Molien Series
F 70.12. MolienSeries
F 70.12. MolienSeries
F 70.12. MolienSeries
F 70.12. MolienSeries
F 70.12. MolienSeriesInfo
F 70.12. ValueMolienSeries
F 70.12. MolienSeriesWithGivenDenominator
S 70.13. Possible Permutation Characters
I 70.13. characters!permutation
I 70.13. candidates!for permutation characters
I 70.13. possible permutation characters
I 70.13. permutation characters!possible
F 70.13. PermCharInfo
F 70.13. PermCharInfo
F 70.13. PermCharInfoRelative
S 70.14. Computing Possible Permutation Characters
I 70.14. characters!permutation
I 70.14. candidates!for permutation characters
I 70.14. possible permutation characters
I 70.14. permutation characters!possible
F 70.14. PermChars
F 70.14. PermChars
F 70.14. PermChars
F 70.14. TestPerm1
F 70.14. TestPerm2
F 70.14. TestPerm3
F 70.14. TestPerm4
F 70.14. TestPerm5
F 70.14. PermBounds
F 70.14. PermComb
F 70.14. Inequalities
S 70.15. Operations for Brauer Characters
F 70.15. FrobeniusCharacterValue
I 70.15. TryConwayPolynomialForFrobeniusCharacterValue
F 70.15. BrauerCharacterValue
F 70.15. SizeOfFieldOfDefinition
F 70.15. RealizableBrauerCharacters
S 70.16. Domains Generated by Class Functions
C ctblmaps.tex 71. Maps Concerning Character Tables
I 71.0. maps
I 71.0. parametrized maps
S 71.1. Power Maps
F 71.1. PowerMap
F 71.1. PowerMapOp
F 71.1. ComputedPowerMaps
F 71.1. PossiblePowerMaps
F 71.1. ElementOrdersPowerMap
F 71.1. PowerMapByComposition
F 71.1. OrbitPowerMaps
I 71.1. matrix automorphisms
F 71.1. RepresentativesPowerMaps
S 71.2. Class Fusions between Character Tables
I 71.2. fusions
I 71.2. subgroup fusions
F 71.2. FusionConjugacyClasses
F 71.2. FusionConjugacyClasses
F 71.2. FusionConjugacyClasses
F 71.2. FusionConjugacyClassesOp
F 71.2. FusionConjugacyClassesOp
F 71.2. ComputedClassFusions
F 71.2. GetFusionMap
F 71.2. GetFusionMap
F 71.2. StoreFusion
F 71.2. NamesOfFusionSources
F 71.2. PossibleClassFusions
F 71.2. OrbitFusions
I 71.2. table automorphisms
F 71.2. RepresentativesFusions
F 71.2. RepresentativesFusions
F 71.2. ConsiderStructureConstants
S 71.3. Parametrized Maps
I 71.3. map!parametrized
I 71.3. class functions
F 71.3. CompositionMaps
F 71.3. InverseMap
F 71.3. ProjectionMap
F 71.3. Indirected
F 71.3. Parametrized
F 71.3. ContainedMaps
F 71.3. UpdateMap
F 71.3. MeetMaps
F 71.3. CommutativeDiagram
F 71.3. CheckFixedPoints
F 71.3. TransferDiagram
F 71.3. TestConsistencyMaps
F 71.3. Indeterminateness
F 71.3. PrintAmbiguity
F 71.3. ContainedSpecialVectors
F 71.3. IntScalarProducts
F 71.3. NonnegIntScalarProducts
F 71.3. ContainedPossibleVirtualCharacters
F 71.3. ContainedPossibleCharacters
I 71.3. IntScalarProducts
I 71.3. NonnegIntScalarProducts
I 71.3. ContainedPossibleVirtualCharacters
I 71.3. ContainedPossibleCharacters
I 71.3. ContainedSpecialVectors
F 71.3. CollapsedMat
F 71.3. ContainedDecomposables
F 71.3. ContainedCharacters
S 71.4. Subroutines for the Construction of Power Maps
F 71.4. InitPowerMap
F 71.4. Congruences!for character tables
F 71.4. ConsiderKernels
F 71.4. ConsiderSmallerPowerMaps
F 71.4. MinusCharacter
F 71.4. PowerMapsAllowedBySymmetrizations
S 71.5. Subroutines for the Construction of Class Fusions
F 71.5. InitFusion
F 71.5. CheckPermChar
I 71.5. permutation character
F 71.5. ConsiderTableAutomorphisms
I 71.5. table automorphisms
F 71.5. FusionsAllowedByRestrictions
C ctblmono.tex 72. Monomiality Questions
F 72.0. InfoMonomial
S 72.1. Character Degrees and Derived Length
F 72.1. Alpha
F 72.1. Delta
F 72.1. IsBergerCondition
F 72.1. IsBergerCondition
S 72.2. Primitivity of Characters
F 72.2. TestHomogeneous
F 72.2. IsPrimitiveCharacter
F 72.2. TestQuasiPrimitive
F 72.2. IsQuasiPrimitive
F 72.2. TestInducedFromNormalSubgroup
F 72.2. IsInducedFromNormalSubgroup
S 72.3. Testing Monomiality
I 72.3. IsMonomial!for groups
I 72.3. IsMonomial!for characters
F 72.3. TestMonomial
F 72.3. TestMonomial
F 72.3. TestMonomial
F 72.3. TestMonomial
F 72.3. TestMonomialUseLattice
F 72.3. IsMonomialNumber
I 72.3. IsMonomial!for positive integers
F 72.3. TestMonomialQuick
F 72.3. TestMonomialQuick
F 72.3. TestSubnormallyMonomial
F 72.3. TestSubnormallyMonomial
F 72.3. IsSubnormallyMonomial
F 72.3. IsSubnormallyMonomial
F 72.3. TestRelativelySM
F 72.3. TestRelativelySM
F 72.3. TestRelativelySM
F 72.3. TestRelativelySM
F 72.3. IsRelativelySM
F 72.3. IsRelativelySM
S 72.4. Minimal Nonmonomial Groups
F 72.4. IsMinimalNonmonomial
F 72.4. MinimalNonmonomialGroup
C install.tex 73. Installing GAP
I 73.0. installation
I 73.0. options
S 73.1. Installation Overview
S 73.2. Get the Archives
S 73.3. Unpacking
S 73.4. Compilation
S 73.5. Test of the installation
S 73.6. Packages
S 73.7. Finish Installation and Cleanup
S 73.8. The Documentation
S 73.9. If Things Go Wrong
I 73.9. problems
I 73.9. FAQ
I 73.9. support!email address
I 73.9. bug reports!see If Things Go Wrong
S 73.10. Known Problems of the Configure Process
S 73.11. Problems on Particular Systems
S 73.12. Optimization and Compiler Options
S 73.13. Porting GAP
S 73.14. GAP for Macintosh OS X
I 73.14. OSX
I 73.14. Apple
I 73.14. Macintosh
S 73.15. GAP for MacOS
I 73.15. MacOS
I 73.15. Apple
I 73.15. Macintosh
S 73.16. Installation of GAP for MacOS
S 73.17. Expert Windows installation
S 73.18. Copyrights
C gappkg.tex 74. GAP Packages
I 74.0. package
S 74.1. Installing a GAP Package
S 74.2. Loading a GAP Package
I 74.2. automatic loading of GAP packages
I 74.2. disable automatic loading
F 74.2. LoadPackage
F 74.2. LoadPackage
I 74.2. NOAUTO
S 74.3. Functions for GAP Packages
F 74.3. ReadPackage
F 74.3. ReadPackage
F 74.3. RereadPackage
F 74.3. RereadPackage
F 74.3. TestPackageAvailability
F 74.3. TestPackageAvailability
F 74.3. InstalledPackageVersion
F 74.3. DirectoriesPackageLibrary
F 74.3. DirectoriesPackagePrograms
F 74.3. CompareVersionNumbers
C obsolete.tex 75. Replaced and Removed Command Names
I 75.0. obsolete
I 75.0. deprecated
I 75.0. legacy
S 75.1. Group Actions - Name Changes
I 75.1. group operations
I 75.1. Operation
I 75.1. RepresentativeOperation
I 75.1. OperationHomomorphism
I 75.1. FunctionOperation
S 75.2. Package Interface - Obsolete Functions and Name Changes
I 75.2. DeclarePackage
I 75.2. DeclareAutoPackage
I 75.2. DeclarePackageDocumentation
I 75.2. DeclarePackageAutoDocumentation
I 75.2. RequirePackage
I 75.2. ReadPkg
I 75.2. RereadPkg
I 75.2. CreateCompletionFilesPkg
S 75.3. Normal Forms of Integer Matrices - Name Changes
I 75.3. Smith normal form
I 75.3. Hermite normal form
S 75.4. Miscellaneous Name Changes or Removed Names
I 75.4. QUIET
I 75.4. BANNER
I 75.4. GAPInfo
I 75.4. MonomialTotalDegreeLess
I 75.4. NormedVectors
C copyrigh.tex 76. Copyright
