GAP Manual

Release 3.4

Martin Schönert together with
Hans Ulrich Besche, Thomas Breuer, Frank Celler,
Volkmar Felsch, Alexander Hulpke, Jürgen Mnich,
Götz Pfeiffer, Udo Polis, Heiko Theissen
(Lehrstuhl D für Mathematik, RWTH Aachen);
Alice Niemeyer
(Mathematics Research Section, ANU, Canberra, Australia).
10. Jul. 1994
(HTML conversion by Martin Schönert)

Welcome to the WWW pages of the GAP Manual archive. Here you find all 63 chapters, all 1278 sections, and the index of the GAP Manual. Comments and suggestions are most welcome.

Table of Chapters

1. About GAP
- 1.1. About Conventions
- 1.2. About Starting and Leaving GAP
- 1.3. About First Steps
- 1.4. About Help
- 1.5. About Syntax Errors
- 1.6. About Constants and Operators
- 1.7. About Variables and Assignments
- 1.8. About Functions
- 1.9. About Lists
- 1.10. About Identical Lists
- 1.11. About Sets
- 1.12. About Vectors and Matrices
- 1.13. About Records
- 1.14. About Ranges
- 1.15. About Loops
- 1.16. About Further List Operations
- 1.17. About Writing Functions
- 1.18. About Groups
- 1.19. About Operations of Groups
- 1.20. About Finitely Presented Groups and Presentations
- 1.21. About Fields
- 1.22. About Matrix Groups
- 1.23. About Domains and Categories
- 1.24. About Mappings and Homomorphisms
- 1.25. About Character Tables
- 1.26. About Group Libraries
- 1.27. About the Implementation of Domains
- 1.28. About Defining New Domains
- 1.29. About Defining New Parametrized Domains
- 1.30. About Defining New Group Elements
2. The Programming Language
- 2.1. Lexical Structure
- 2.2. Symbols
- 2.3. Whitespaces
- 2.4. Keywords
- 2.5. Identifiers
- 2.6. Expressions
- 2.7. Variables
- 2.8. Function Calls
- 2.9. Comparisons
- 2.10. Operations
- 2.11. Statements
- 2.12. Assignments
- 2.13. Procedure Calls
- 2.14. If
- 2.15. While
- 2.16. Repeat
- 2.17. For
- 2.18. Functions
- 2.19. Return
- 2.20. The Syntax in BNF
3. Environment
- 3.1. Main Loop
- 3.2. Break Loops
- 3.3. Error
- 3.4. Line Editing
- 3.5. Help
- 3.6. Reading Sections
- 3.7. Format of Sections
- 3.8. Browsing through the Sections
- 3.9. Redisplaying a Section
- 3.10. Abbreviating Section Names
- 3.11. Help Index
- 3.12. Read
- 3.13. ReadLib
- 3.14. Print
- 3.15. PrintTo
- 3.16. AppendTo
- 3.17. LogTo
- 3.18. LogInputTo
- 3.19. SizeScreen
- 3.20. Runtime
- 3.21. Profile
- 3.22. Exec
- 3.23. Edit
4. Domains
- 4.1. Domain Records
- 4.2. Dispatchers
- 4.3. More about Dispatchers
- 4.4. An Example of a Computation in a Domain
- 4.5. Domain
- 4.6. Elements
- 4.7. Comparisons of Domains
- 4.8. Membership Test for Domains
- 4.9. IsFinite
- 4.10. Size
- 4.11. IsSubset
- 4.12. Intersection
- 4.13. Union
- 4.14. Difference
- 4.15. Representative
- 4.16. Random
5. Rings
- 5.1. IsRing
- 5.2. Ring
- 5.3. DefaultRing
- 5.4. Comparisons of Ring Elements
- 5.5. Operations for Ring Elements
- 5.6. Quotient
- 5.7. IsCommutativeRing
- 5.8. IsIntegralRing
- 5.9. IsUniqueFactorizationRing
- 5.10. IsEuclideanRing
- 5.11. IsUnit
- 5.12. Units
- 5.13. IsAssociated
- 5.14. StandardAssociate
- 5.15. Associates
- 5.16. IsIrreducible
- 5.17. IsPrime
- 5.18. Factors
- 5.19. EuclideanDegree
- 5.20. EuclideanRemainder
- 5.21. EuclideanQuotient
- 5.22. QuotientRemainder
- 5.23. Mod
- 5.24. QuotientMod
- 5.25. PowerMod
- 5.26. Gcd
- 5.27. GcdRepresentation
- 5.28. Lcm
- 5.29. Ring Records
6. Fields
- 6.1. IsField
- 6.2. Field
- 6.3. DefaultField
- 6.4. Fields over Subfields
- 6.5. Comparisons of Field Elements
- 6.6. Operations for Field Elements
- 6.7. GaloisGroup
- 6.8. MinPol
- 6.9. CharPol
- 6.10. Norm
- 6.11. Trace
- 6.12. Conjugates
- 6.13. Field Homomorphisms
- 6.14. IsFieldHomomorphism
- 6.15. KernelFieldHomomorphism
- 6.16. Mapping Functions for Field Homomorphisms
- 6.17. Field Records
7. Groups
- 7.1. Group Elements
- 7.2. Comparisons of Group Elements
- 7.3. Operations for Group Elements
- 7.4. IsGroupElement
- 7.5. Order
- 7.6. More about Groups and Subgroups
- 7.7. IsParent
- 7.8. Parent
- 7.9. Group
- 7.10. AsGroup
- 7.11. IsGroup
- 7.12. Subgroup
- 7.13. AsSubgroup
- 7.14. Subgroups
- 7.15. Agemo
- 7.16. Centralizer
- 7.17. Centre
- 7.18. Closure
- 7.19. CommutatorSubgroup
- 7.20. ConjugateSubgroup
- 7.21. Core
- 7.22. DerivedSubgroup
- 7.23. FittingSubgroup
- 7.24. FrattiniSubgroup
- 7.25. NormalClosure
- 7.26. NormalIntersection
- 7.27. Normalizer
- 7.28. PCore
- 7.29. PrefrattiniSubgroup
- 7.30. Radical
- 7.31. SylowSubgroup
- 7.32. TrivialSubgroup
- 7.33. FactorGroup
- 7.34. FactorGroupElement
- 7.35. CommutatorFactorGroup
- 7.36. Series of Subgroups
- 7.37. DerivedSeries
- 7.38. CompositionSeries
- 7.39. ElementaryAbelianSeries
- 7.40. JenningsSeries
- 7.41. LowerCentralSeries
- 7.42. PCentralSeries
- 7.43. SubnormalSeries
- 7.44. UpperCentralSeries
- 7.45. Properties and Property Tests
- 7.46. AbelianInvariants
- 7.47. DimensionsLoewyFactors
- 7.48. EulerianFunction
- 7.49. Exponent
- 7.50. Factorization
- 7.51. Index
- 7.52. IsAbelian
- 7.53. IsCentral
- 7.54. IsConjugate
- 7.55. IsCyclic
- 7.56. IsElementaryAbelian
- 7.57. IsNilpotent
- 7.58. IsNormal
- 7.59. IsPerfect
- 7.60. IsSimple
- 7.61. IsSolvable
- 7.62. IsSubgroup
- 7.63. IsSubnormal
- 7.64. IsTrivial for Groups
- 7.65. GroupId
- 7.66. PermutationCharacter
- 7.67. Conjugacy Classes
- 7.68. ConjugacyClasses
- 7.69. ConjugacyClass
- 7.70. IsConjugacyClass
- 7.71. Set Functions for Conjugacy Classes
- 7.72. Conjugacy Class Records
- 7.73. ConjugacyClassesSubgroups
- 7.74. Lattice
- 7.75. ConjugacyClassSubgroups
- 7.76. IsConjugacyClassSubgroups
- 7.77. Set Functions for Subgroup Conjugacy Classes
- 7.78. Subgroup Conjugacy Class Records
- 7.79. ConjugacyClassesMaximalSubgroups
- 7.80. MaximalSubgroups
- 7.81. NormalSubgroups
- 7.82. ConjugateSubgroups
- 7.83. Cosets of Subgroups
- 7.84. RightCosets
- 7.85. RightCoset
- 7.86. IsRightCoset
- 7.87. Set Functions for Right Cosets
- 7.88. Right Cosets Records
- 7.89. LeftCosets
- 7.90. LeftCoset
- 7.91. IsLeftCoset
- 7.92. DoubleCosets
- 7.93. DoubleCoset
- 7.94. IsDoubleCoset
- 7.95. Set Functions for Double Cosets
- 7.96. Double Coset Records
- 7.97. Group Constructions
- 7.98. DirectProduct
- 7.99. DirectProduct for Groups
- 7.100. SemidirectProduct
- 7.101. SemidirectProduct for Groups
- 7.102. SubdirectProduct
- 7.103. WreathProduct
- 7.104. WreathProduct for Groups
- 7.105. Group Homomorphisms
- 7.106. IsGroupHomomorphism
- 7.107. KernelGroupHomomorphism
- 7.108. Mapping Functions for Group Homomorphisms
- 7.109. NaturalHomomorphism
- 7.110. ConjugationGroupHomomorphism
- 7.111. InnerAutomorphism
- 7.112. GroupHomomorphismByImages
- 7.113. Set Functions for Groups
- 7.114. Elements for Groups
- 7.115. Intersection for Groups
- 7.116. Operations for Groups
- 7.117. Group Records
8. Operations of Groups
- 8.1. Other Operations
- 8.2. Cycle
- 8.3. CycleLength
- 8.4. Cycles
- 8.5. CycleLengths
- 8.6. Permutation
- 8.7. IsFixpoint
- 8.8. IsFixpointFree
- 8.9. DegreeOperation
- 8.10. IsTransitive
- 8.11. Transitivity
- 8.12. IsRegular
- 8.13. IsSemiRegular
- 8.14. Orbit
- 8.15. OrbitLength
- 8.16. Orbits
- 8.17. OrbitLengths
- 8.18. Operation
- 8.19. OperationHomomorphism
- 8.20. Blocks
- 8.21. IsPrimitive
- 8.22. Stabilizer
- 8.23. RepresentativeOperation
- 8.24. RepresentativesOperation
- 8.25. IsEquivalentOperation
9. Vector Spaces
- 9.1. VectorSpace
- 9.2. IsVectorSpace
- 9.3. Vector Space Records
- 9.4. Set Functions for Vector Spaces
- 9.5. IsSubspace
- 9.6. Base
- 9.7. AddBase
- 9.8. Dimension
- 9.9. LinearCombination
- 9.10. Coefficients
10. Integers
- 10.1. Comparisons of Integers
- 10.2. Operations for Integers
- 10.3. QuoInt
- 10.4. RemInt
- 10.5. IsInt
- 10.6. Int
- 10.7. AbsInt
- 10.8. SignInt
- 10.9. ChineseRem
- 10.10. LogInt
- 10.11. RootInt
- 10.12. SmallestRootInt
- 10.13. Set Functions for Integers
- 10.14. Ring Functions for Integers
- 10.15. Primes
- 10.16. IsPrimeInt
- 10.17. IsPrimePowerInt
- 10.18. NextPrimeInt
- 10.19. PrevPrimeInt
- 10.20. FactorsInt
- 10.21. DivisorsInt
- 10.22. Sigma
- 10.23. Tau
- 10.24. MoebiusMu
11. Number Theory
- 11.1. PrimeResidues
- 11.2. Phi
- 11.3. Lambda
- 11.4. OrderMod
- 11.5. IsPrimitiveRootMod
- 11.6. PrimitiveRootMod
- 11.7. Jacobi
- 11.8. Legendre
- 11.9. RootMod
- 11.10. RootsUnityMod
12. Rationals
- 12.1. IsRat
- 12.2. Numerator
- 12.3. Denominator
- 12.4. Comparisons of Rationals
- 12.5. Operations for Rationals
- 12.6. Set Functions for Rationals
- 12.7. Field Functions for Rationals
13. Cyclotomics
- 13.1. More about Cyclotomics
- 13.2. Cyclotomic Integers
- 13.3. IntCyc
- 13.4. RoundCyc
- 13.5. IsCyc
- 13.6. IsCycInt
- 13.7. NofCyc
- 13.8. CoeffsCyc
- 13.9. Comparisons of Cyclotomics
- 13.10. Operations for Cyclotomics
- 13.11. GaloisCyc
- 13.12. ATLAS irrationalities
- 13.13. StarCyc
- 13.14. Quadratic
- 13.15. GaloisMat
- 13.16. RationalizedMat
14. Gaussians
- 14.1. Comparisons of Gaussians
- 14.2. Operations for Gaussians
- 14.3. IsGaussRat
- 14.4. IsGaussInt
- 14.5. Set Functions for Gaussians
- 14.6. Field Functions for Gaussian Rationals
- 14.7. Ring Functions for Gaussian Integers
- 14.8. TwoSquares
15. Subfields of Cyclotomic Fields
- 15.1. IsNumberField
- 15.2. IsCyclotomicField
- 15.3. Number Field Records
- 15.4. Cyclotomic Field Records
- 15.5. DefaultField and Field for Cyclotomics
- 15.6. DefaultRing and Ring for Cyclotomic Integers
- 15.7. GeneratorsPrimeResidues
- 15.8. GaloisGroup for Number Fields
- 15.9. ZumbroichBase
- 15.10. Integral Bases for Number Fields
- 15.11. NormalBaseNumberField
- 15.12. Coefficients for Number Fields
- 15.13. Domain Functions for Number Fields
16. Algebraic extensions of fields
- 16.1. AlgebraicExtension
- 16.2. IsAlgebraicExtension
- 16.3. RootOf
- 16.4. Algebraic Extension Elements
- 16.5. Set functions for Algebraic Extensions
- 16.6. IsNormalExtension
- 16.7. MinpolFactors
- 16.8. GaloisGroup for Extension Fields
- 16.9. ExtensionAutomorphism
- 16.10. Field functions for Algebraic Extensions
- 16.11. Algebraic Extension Records
- 16.12. Extension Element Records
- 16.13. IsAlgebraicElement
- 16.14. Algebraic extensions of the Rationals
- 16.15. DefectApproximation
- 16.16. GaloisType
- 16.17. ProbabilityShapes
- 16.18. DecomPoly
17. Unknowns
- 17.1. Unknown
- 17.2. IsUnknown
- 17.3. Comparisons of Unknowns
- 17.4. Operations for Unknowns
18. Finite Fields
- 18.1. Finite Field Elements
- 18.2. Comparisons of Finite Field Elements
- 18.3. Operations for Finite Field Elements
- 18.4. IsFFE
- 18.5. CharFFE
- 18.6. DegreeFFE
- 18.7. OrderFFE
- 18.8. IntFFE
- 18.9. LogFFE
- 18.10. GaloisField
- 18.11. FrobeniusAutomorphism
- 18.12. Set Functions for Finite Fields
- 18.13. Field Functions for Finite Fields
19. Polynomials
- 19.1. Multivariate Polynomials
- 19.2. Indeterminate
- 19.3. Polynomial
- 19.4. IsPolynomial
- 19.5. Comparisons of Polynomials
- 19.6. Operations for Polynomials
- 19.7. Degree
- 19.8. LeadingCoefficient
- 19.9. Value
- 19.10. Derivative
- 19.11. InterpolatedPolynomial
- 19.12. ConwayPolynomial
- 19.13. CyclotomicPolynomial
- 19.14. PolynomialRing
- 19.15. IsPolynomialRing
- 19.16. LaurentPolynomialRing
- 19.17. IsLaurentPolynomialRing
- 19.18. Ring Functions for Polynomial Rings
- 19.19. Ring Functions for Laurent Polynomial Rings
20. Permutations
- 20.1. Comparisons of Permutations
- 20.2. Operations for Permutations
- 20.3. IsPerm
- 20.4. LargestMovedPointPerm
- 20.5. SmallestMovedPointPerm
- 20.6. SignPerm
- 20.7. SmallestGeneratorPerm
- 20.8. ListPerm
- 20.9. PermList
- 20.10. RestrictedPerm
- 20.11. MappingPermListList
21. Permutation Groups
- 21.1. IsPermGroup
- 21.2. PermGroupOps.MovedPoints
- 21.3. PermGroupOps.SmallestMovedPoint
- 21.4. PermGroupOps.LargestMovedPoint
- 21.5. PermGroupOps.NrMovedPoints
- 21.6. Stabilizer Chains
- 21.7. StabChain
- 21.8. MakeStabChain
- 21.9. ExtendStabChain
- 21.10. ReduceStabChain
- 21.11. MakeStabChainStrongGenerators
- 21.12. Base for Permutation Groups
- 21.13. PermGroupOps.Indices
- 21.14. PermGroupOps.StrongGenerators
- 21.15. ListStabChain
- 21.16. PermGroupOps.ElementProperty
- 21.17. PermGroupOps.SubgroupProperty
- 21.18. CentralCompositionSeriesPPermGroup
- 21.19. PermGroupOps.PgGroup
- 21.20. Set Functions for Permutation Groups
- 21.21. Group Functions for Permutation Groups
- 21.22. Operations of Permutation Groups
- 21.23. Homomorphisms for Permutation Groups
- 21.24. Random Methods for Permutation Groups
- 21.25. Permutation Group Records
22. Words in Abstract Generators
- 22.1. AbstractGenerator
- 22.2. AbstractGenerators
- 22.3. Comparisons of Words
- 22.4. Operations for Words
- 22.5. IsWord
- 22.6. LengthWord
- 22.7. ExponentSumWord
- 22.8. Subword
- 22.9. PositionWord
- 22.10. SubstitutedWord
- 22.11. EliminatedWord
- 22.12. MappedWord
23. Finitely Presented Groups
- 23.1. FreeGroup
- 23.2. Set Functions for Finitely Presented Groups
- 23.3. Group Functions for Finitely Presented Groups
- 23.4. CosetTableFpGroup
- 23.5. OperationCosetsFpGroup
- 23.6. IsIdenticalPresentationFpGroup
- 23.7. LowIndexSubgroupsFpGroup
- 23.8. Presentation Records
- 23.9. Changing Presentations
- 23.10. Subgroup Presentations
- 23.11. SimplifiedFpGroup
- 23.12. Tietze Transformations
- 23.13. DecodeTree
24. Words in Finite Polycyclic Groups
- 24.1. More about Ag Words
- 24.2. Ag Word Comparisons
- 24.3. CentralWeight
- 24.4. CompositionLength
- 24.5. Depth
- 24.6. IsAgWord
- 24.7. LeadingExponent
- 24.8. RelativeOrder
- 24.9. CanonicalAgWord
- 24.10. DifferenceAgWord
- 24.11. ReducedAgWord
- 24.12. SiftedAgWord
- 24.13. SumAgWord
- 24.14. ExponentAgWord
- 24.15. ExponentsAgWord
25. Finite Polycyclic Groups
- 25.1. More about Ag Groups
- 25.2. Construction of Ag Groups
- 25.3. Ag Group Operations
- 25.4. Ag Group Records
- 25.5. Set Functions for Ag Groups
- 25.6. Elements for Ag Groups
- 25.7. Intersection for Ag Groups
- 25.8. Size for Ag Groups
- 25.9. Group Functions for Ag Groups
- 25.10. AsGroup for Ag Groups
- 25.11. Group for Ag Groups
- 25.12. CommutatorSubgroup for Ag Groups
- 25.13. Normalizer for Ag Groups
- 25.14. AbelianInvariants for Ag Groups
- 25.15. IsCyclic for Ag Groups
- 25.16. IsNormal for Ag Groups
- 25.17. IsSubgroup for Ag Groups
- 25.18. Stabilizer for Ag Groups
- 25.19. CyclicGroup for Ag Groups
- 25.20. ElementaryAbelianGroup for Ag Groups
- 25.21. DirectProduct for Ag Groups
- 25.22. WreathProduct for Ag Groups
- 25.23. RightCoset for Ag Groups
- 25.24. FpGroup for Ag Groups
- 25.25. Ag Group Functions
- 25.26. AgGroup
- 25.27. IsAgGroup
- 25.28. AgGroupFpGroup
- 25.29. IsConsistent
- 25.30. IsElementaryAbelianAgSeries
- 25.31. MatGroupAgGroup
- 25.32. PermGroupAgGroup
- 25.33. RefinedAgSeries
- 25.34. ChangeCollector
- 25.35. The Prime Quotient Algorithm
- 25.36. PQuotient
- 25.37. Save
- 25.38. PQp
- 25.39. InitPQp
- 25.40. FirstClassPQp
- 25.41. NextClassPQp
- 25.42. Weight
- 25.43. Factorization for PQp
- 25.44. The Solvable Quotient Algorithm
- 25.45. SolvableQuotient
- 25.46. InitSQ
- 25.47. ModulesSQ
- 25.48. NextModuleSQ
- 25.49. Generating Systems of Ag Groups
- 25.50. AgSubgroup
- 25.51. Cgs
- 25.52. Igs
- 25.53. IsNormalized
- 25.54. Normalize
- 25.55. Normalized
- 25.56. MergedCgs
- 25.57. MergedIgs
- 25.58. Factor Groups of Ag Groups
- 25.59. FactorGroup for AgGroups
- 25.60. CollectorlessFactorGroup
- 25.61. FactorArg
- 25.62. Subgroups and Properties of Ag Groups
- 25.63. CompositionSubgroup
- 25.64. HallSubgroup
- 25.65. PRump
- 25.66. RefinedSubnormalSeries
- 25.67. SylowComplements
- 25.68. SylowSystem
- 25.69. SystemNormalizer
- 25.70. MinimalGeneratingSet
- 25.71. IsElementAgSeries
- 25.72. IsPNilpotent
- 25.73. NumberConjugacyClasses
- 25.74. Exponents
- 25.75. FactorsAgGroup
- 25.76. MaximalElement
- 25.77. Orbitalgorithms of Ag Groups
- 25.78. AffineOperation
- 25.79. AgOrbitStabilizer
- 25.80. LinearOperation
- 25.81. Intersections of Ag Groups
- 25.82. ExtendedIntersectionSumAgGroup
- 25.83. IntersectionSumAgGroup
- 25.84. SumAgGroup
- 25.85. SumFactorizationFunctionAgGroup
- 25.86. One Cohomology Group
- 25.87. OneCoboundaries
- 25.88. OneCocycles
- 25.89. Complements
- 25.90. Complement
- 25.91. Complementclasses
- 25.92. CoprimeComplement
- 25.93. ComplementConjugatingAgWord
- 25.94. HallConjugatingWordAgGroup
- 25.95. Example, normal closure
26. Special Ag Groups
- 26.1. More about Special Ag Groups
- 26.2. Construction of Special Ag Groups
- 26.3. Restricted Special Ag Groups
- 26.4. Special Ag Group Records
- 26.5. MatGroupSagGroup
- 26.6. DualMatGroupSagGroup
- 26.7. Ag Group Functions for Special Ag Groups
27. Lists
- 27.1. IsList
- 27.2. List
- 27.3. List Elements
- 27.4. Length
- 27.5. List Assignment
- 27.6. Add
- 27.7. Append
- 27.8. Identical Lists
- 27.9. IsIdentical
- 27.10. Enlarging Lists
- 27.11. Comparisons of Lists
- 27.12. Operations for Lists
- 27.13. In
- 27.14. Position
- 27.15. PositionSorted
- 27.16. PositionProperty
- 27.17. Concatenation
- 27.18. Flat
- 27.19. Reversed
- 27.20. Sublist
- 27.21. Cartesian
- 27.22. Number
- 27.23. Collected
- 27.24. Filtered
- 27.25. ForAll
- 27.26. ForAny
- 27.27. First
- 27.28. Sort
- 27.29. SortParallel
- 27.30. Sortex
- 27.31. Permuted
- 27.32. Product
- 27.33. Sum
- 27.34. Maximum
- 27.35. Minimum
- 27.36. Iterated
- 27.37. RandomList
28. Sets
- 28.1. IsSet
- 28.2. Set
- 28.3. IsEqualSet
- 28.4. AddSet
- 28.5. RemoveSet
- 28.6. UniteSet
- 28.7. IntersectSet
- 28.8. SubtractSet
- 28.9. Set Functions for Sets
- 28.10. More about Sets
29. Boolean Lists
- 29.1. BlistList
- 29.2. ListBlist
- 29.3. IsBlist
- 29.4. SizeBlist
- 29.5. IsSubsetBlist
- 29.6. UnionBlist
- 29.7. IntersectionBlist
- 29.8. DifferenceBlist
- 29.9. UniteBlist
- 29.10. IntersectBlist
- 29.11. SubtractBlist
- 29.12. More about Boolean Lists
30. Strings and Characters
- 30.1. String
- 30.2. ConcatenationString
- 30.3. SubString
- 30.4. Comparisons of Strings
- 30.5. IsString
- 30.6. LengthString
31. Ranges
- 31.1. IsRange
- 31.2. More about Ranges
32. Vectors
- 32.1. Operations for Vectors
- 32.2. IsVector
- 32.3. NormedVector
- 32.4. More about Vectors
33. Row Spaces
- 33.1. More about Row Spaces
- 33.2. Row Space Bases
- 33.3. Row Space Cosets
- 33.4. Quotient Spaces
- 33.5. Subspaces and Parent Spaces
- 33.6. RowSpace
- 33.7. Operations for Row Spaces
- 33.8. Functions for Row Spaces
- 33.9. IsRowSpace
- 33.10. Subspace
- 33.11. AsSubspace
- 33.12. AsSpace
- 33.13. NormedVectors
- 33.14. Coefficients for Row Space Bases
- 33.15. SiftedVector
- 33.16. Basis
- 33.17. CanonicalBasis
- 33.18. SemiEchelonBasis
- 33.19. IsSemiEchelonBasis
- 33.20. NumberVector
- 33.21. ElementRowSpace
- 33.22. Operations for Row Space Cosets
- 33.23. Functions for Row Space Cosets
- 33.24. IsSpaceCoset
- 33.25. Operations for Quotient Spaces
- 33.26. Functions for Quotient Spaces
- 33.27. Row Space Records
- 33.28. Row Space Basis Records
- 33.29. Row Space Coset Records
- 33.30. Quotient Space Records
34. Matrices
- 34.1. Operations for Matrices
- 34.2. IsMat
- 34.3. IdentityMat
- 34.4. NullMat
- 34.5. TransposedMat
- 34.6. KroneckerProduct
- 34.7. DimensionsMat
- 34.8. TraceMat
- 34.9. DeterminantMat
- 34.10. RankMat
- 34.11. OrderMat
- 34.12. TriangulizeMat
- 34.13. BaseMat
- 34.14. NullspaceMat
- 34.15. SolutionMat
- 34.16. DiagonalizeMat
- 34.17. ElementaryDivisorsMat
- 34.18. PrintArray
35. Matrix Rings
- 35.1. Set Functions for Matrix Rings
- 35.2. Ring Functions for Matrix Rings
36. Matrix Groups
- 36.1. Set Functions for Matrix Groups
- 36.2. Group Functions for Matrix Groups
- 36.3. Matrix Group Records
37. Group Libraries
- 37.1. The Basic Groups Library
- 37.2. Selection Functions
- 37.3. Example Functions
- 37.4. Extraction Functions
- 37.5. The Primitive Groups Library
- 37.6. The Transitive Groups Library
- 37.7. The Solvable Groups Library
- 37.8. The 2-Groups Library
- 37.9. The 3-Groups Library
- 37.10. The Irreducible Solvable Linear Groups Library
- 37.11. The Crystallographic Groups Library
38. Algebras
- 38.1. More about Algebras
- 38.2. Algebras and Unital Algebras
- 38.3. Parent Algebras and Subalgebras
- 38.4. Algebra
- 38.5. UnitalAlgebra
- 38.6. IsAlgebra
- 38.7. IsUnitalAlgebra
- 38.8. Subalgebra
- 38.9. UnitalSubalgebra
- 38.10. IsSubalgebra
- 38.11. AsAlgebra
- 38.12. AsUnitalAlgebra
- 38.13. AsSubalgebra
- 38.14. AsUnitalSubalgebra
- 38.15. Operations for Algebras
- 38.16. Zero and One for Algebras
- 38.17. Set Theoretic Functions for Algebras
- 38.18. Property Tests for Algebras
- 38.19. Vector Space Functions for Algebras
- 38.20. Algebra Functions for Algebras
- 38.21. TrivialSubalgebra
- 38.22. Operation for Algebras
- 38.23. OperationHomomorphism for Algebras
- 38.24. Algebra Homomorphisms
- 38.25. Mapping Functions for Algebra Homomorphisms
- 38.26. Algebra Elements
- 38.27. IsAlgebraElement
- 38.28. Algebra Records
- 38.29. FFList
39. Finitely Presented Algebras
- 39.1. More about Finitely Presented Algebras
- 39.2. FreeAlgebra
- 39.3. FpAlgebra
- 39.4. IsFpAlgebra
- 39.5. Operators for Finitely Presented Algebras
- 39.6. Functions for Finitely Presented Algebras
- 39.7. PrintDefinitionFpAlgebra
- 39.8. MappedExpression
- 39.9. Elements of Finitely Presented Algebras
- 39.10. ElementAlgebra
- 39.11. NumberAlgebraElement
40. Matrix Algebras
- 40.1. More about Matrix Algebras
- 40.2. Bases for Matrix Algebras
- 40.3. IsMatAlgebra
- 40.4. Zero and One for Matrix Algebras
- 40.5. Functions for Matrix Algebras
- 40.6. Algebra Functions for Matrix Algebras
- 40.7. RepresentativeOperation for Matrix Algebras
- 40.8. MatAlgebra
- 40.9. NullAlgebra
- 40.10. Fingerprint
- 40.11. NaturalModule
41. Modules
- 41.1. More about Modules
- 41.2. Row Modules
- 41.3. Free Modules
- 41.4. Module
- 41.5. Submodule
- 41.6. AsModule
- 41.7. AsSubmodule
- 41.8. AsSpace for Modules
- 41.9. IsModule
- 41.10. IsFreeModule
- 41.11. Operations for Row Modules
- 41.12. Functions for Row Modules
- 41.13. StandardBasis for Row Modules
- 41.14. IsEquivalent for Row Modules
- 41.15. IsIrreducible for Row Modules
- 41.16. FixedSubmodule
- 41.17. Module Homomorphisms
- 41.18. Row Module Records
- 41.19. Module Homomorphism Records
42. Mappings
- 42.1. IsGeneralMapping
- 42.2. IsMapping
- 42.3. IsInjective
- 42.4. IsSurjective
- 42.5. IsBijection
- 42.6. Comparisons of Mappings
- 42.7. Operations for Mappings
- 42.8. Image
- 42.9. Images
- 42.10. ImagesRepresentative
- 42.11. PreImage
- 42.12. PreImages
- 42.13. PreImagesRepresentative
- 42.14. CompositionMapping
- 42.15. PowerMapping
- 42.16. InverseMapping
- 42.17. IdentityMapping
- 42.18. MappingByFunction
- 42.19. Mapping Records
43. Homomorphisms
- 43.1. IsHomomorphism
- 43.2. IsMonomorphism
- 43.3. IsEpimorphism
- 43.4. IsIsomorphism
- 43.5. IsEndomorphism
- 43.6. IsAutomorphism
- 43.7. Kernel
44. Booleans
- 44.1. Comparisons of Booleans
- 44.2. Operations for Booleans
- 44.3. IsBool
45. Records
- 45.1. Accessing Record Elements
- 45.2. Record Assignment
- 45.3. Identical Records
- 45.4. Comparisons of Records
- 45.5. Operations for Records
- 45.6. In for Records
- 45.7. Printing of Records
- 45.8. IsRec
- 45.9. IsBound
- 45.10. Unbind
- 45.11. Copy
- 45.12. ShallowCopy
- 45.13. RecFields
46. Combinatorics
- 46.1. Factorial
- 46.2. Binomial
- 46.3. Bell
- 46.4. Stirling1
- 46.5. Stirling2
- 46.6. Combinations
- 46.7. Arrangements
- 46.8. UnorderedTuples
- 46.9. Tuples
- 46.10. PermutationsList
- 46.11. Derangements
- 46.12. Permanent
- 46.13. PartitionsSet
- 46.14. Partitions
- 46.15. OrderedPartitions
- 46.16. RestrictedPartitions
- 46.17. SignPartition
- 46.18. AssociatedPartition
- 46.19. PowerPartition
- 46.20. PartitionTuples
- 46.21. Fibonacci
- 46.22. Lucas
- 46.23. Bernoulli
47. Tables of Marks
- 47.1. More about Tables of Marks
- 47.2. Table of Marks Records
- 47.3. The Library of Tables of Marks
- 47.4. TableOfMarks
- 47.5. Marks
- 47.6. NrSubs
- 47.7. WeightsTom
- 47.8. MatTom
- 47.9. TomMat
- 47.10. DecomposedFixedPointVector
- 47.11. TestTom
- 47.12. DisplayTom
- 47.13. NormalizerTom
- 47.14. IntersectionsTom
- 47.15. IsCyclicTom
- 47.16. FusionCharTableTom
- 47.17. PermCharsTom
- 47.18. MoebiusTom
- 47.19. CyclicExtensionsTom
- 47.20. IdempotentsTom
- 47.21. ClassTypesTom
- 47.22. ClassNamesTom
- 47.23. TomCyclic
- 47.24. TomDihedral
- 47.25. TomFrobenius
48. Character Tables
- 48.1. Some Notes on Character Theory in GAP
- 48.2. Character Table Records
- 48.3. Brauer Table Records
- 48.4. IsCharTable
- 48.5. PrintCharTable
- 48.6. TestCharTable
- 48.7. Operations Records for Character Tables
- 48.8. Functions for Character Tables
- 48.9. Operators for Character Tables
- 48.10. Conventions for Character Tables
- 48.11. Getting Character Tables
- 48.12. CharTable
- 48.13. Advanced Methods for Dixon Schneider Calculations
- 48.14. An Example of Advanced Dixon Schneider Calculations
- 48.15. CharTableFactorGroup
- 48.16. CharTableNormalSubgroup
- 48.17. CharTableDirectProduct
- 48.18. CharTableWreathSymmetric
- 48.19. CharTableRegular
- 48.20. CharTableIsoclinic
- 48.21. CharTableSplitClasses
- 48.22. CharTableCollapsedClasses
- 48.23. CharDegAgGroup
- 48.24. CharTableSSGroup
- 48.25. MatRepresentationsPGroup
- 48.26. CharTablePGroup
- 48.27. InitClassesCharTable
- 48.28. InverseClassesCharTable
- 48.29. ClassNamesCharTable
- 48.30. ClassMultCoeffCharTable
- 48.31. MatClassMultCoeffsCharTable
- 48.32. ClassStructureCharTable
- 48.33. RealClassesCharTable
- 48.34. ClassOrbitCharTable
- 48.35. ClassRootsCharTable
- 48.36. NrPolyhedralSubgroups
- 48.37. DisplayCharTable
- 48.38. SortCharactersCharTable
- 48.39. SortClassesCharTable
- 48.40. SortCharTable
- 48.41. MatAutomorphisms
- 48.42. TableAutomorphisms
- 48.43. TransformingPermutations
- 48.44. TransformingPermutationsCharTables
- 48.45. GetFusionMap
- 48.46. StoreFusion
- 48.47. FusionConjugacyClasses
- 48.48. MAKElb11
- 48.49. ScanMOC
- 48.50. MOCChars
- 48.51. GAPChars
- 48.52. MOCTable
- 48.53. PrintToMOC
- 48.54. PrintToCAS
49. Generic Character Tables
- 49.1. More about Generic Character Tables
- 49.2. Examples of Generic Character Tables
- 49.3. CharTableSpecialized
50. Characters
- 50.1. ScalarProduct
- 50.2. MatScalarProducts
- 50.3. Decomposition
- 50.4. Subroutines of Decomposition
- 50.5. KernelChar
- 50.6. PrimeBlocks
- 50.7. Indicator
- 50.8. Eigenvalues
- 50.9. MolienSeries
- 50.10. Reduced
- 50.11. ReducedOrdinary
- 50.12. Tensored
- 50.13. Symmetrisations
- 50.14. SymmetricParts
- 50.15. AntiSymmetricParts
- 50.16. MinusCharacter
- 50.17. OrthogonalComponents
- 50.18. SymplecticComponents
- 50.19. IrreducibleDifferences
- 50.20. Restricted
- 50.21. Inflated
- 50.22. Induced
- 50.23. InducedCyclic
- 50.24. CollapsedMat
- 50.25. Power
- 50.26. Permutation Character Candidates
- 50.27. IsPermChar
- 50.28. PermCharInfo
- 50.29. Inequalities
- 50.30. PermBounds
- 50.31. PermChars
- 50.32. Faithful Permutation Characters
- 50.33. LLLReducedBasis
- 50.34. LLLReducedGramMat
- 50.35. LLL
- 50.36. OrthogonalEmbeddings
- 50.37. ShortestVectors
- 50.38. Extract
- 50.39. Decreased
- 50.40. DnLattice
- 50.41. ContainedDecomposables
- 50.42. ContainedCharacters
- 50.43. ContainedSpecialVectors
- 50.44. ContainedPossibleCharacters
- 50.45. ContainedPossibleVirtualCharacters
51. Maps and Parametrized Maps
- 51.1. More about Maps and Parametrized Maps
- 51.2. CompositionMaps
- 51.3. InverseMap
- 51.4. ProjectionMap
- 51.5. Parametrized
- 51.6. ContainedMaps
- 51.7. UpdateMap
- 51.8. CommutativeDiagram
- 51.9. TransferDiagram
- 51.10. Indeterminateness
- 51.11. PrintAmbiguity
- 51.12. Powermap
- 51.13. SubgroupFusions
- 51.14. InitPowermap
- 51.15. Congruences
- 51.16. ConsiderKernels
- 51.17. ConsiderSmallerPowermaps
- 51.18. InitFusion
- 51.19. CheckPermChar
- 51.20. CheckFixedPoints
- 51.21. TestConsistencyMaps
- 51.22. ConsiderTableAutomorphisms
- 51.23. PowermapsAllowedBySymmetrisations
- 51.24. FusionsAllowedByRestrictions
- 51.25. OrbitFusions
- 51.26. OrbitPowermaps
- 51.27. RepresentativesFusions
- 51.28. RepresentativesPowermaps
- 51.29. Indirected
- 51.30. Powmap
- 51.31. ElementOrdersPowermap
52. Character Table Libraries
- 52.1. Contents of the Table Libraries
- 52.2. Selecting Library Tables
- 52.3. ATLAS Tables
- 52.4. Examples of the ATLAS format for GAP tables
- 52.5. CAS Tables
- 52.6. Organization of the Table Libraries
- 52.7. How to Extend a Table Library
- 52.8. FirstNameCharTable
- 52.9. FileNameCharTable
53. Class Functions
- 53.1. Why Group Characters
- 53.2. More about Class Functions
- 53.3. Operators for Class Functions
- 53.4. Functions for Class Functions
- 53.5. ClassFunction
- 53.6. VirtualCharacter
- 53.7. Character
- 53.8. IsClassFunction
- 53.9. IsVirtualCharacter
- 53.10. IsCharacter
- 53.11. Irr
- 53.12. InertiaSubgroup
- 53.13. OrbitsCharacters
- 53.14. Storing Subgroup Information
- 53.15. NormalSubgroupClasses
- 53.16. ClassesNormalSubgroup
- 53.17. FactorGroupNormalSubgroupClasses
- 53.18. Class Function Records
54. Monomiality Questions
- 54.1. More about Monomiality Questions
- 54.2. Alpha
- 54.3. Delta
- 54.4. BergerCondition
- 54.5. TestHomogeneous
- 54.6. TestQuasiPrimitive
- 54.7. IsPrimitive for Characters
- 54.8. TestInducedFromNormalSubgroup
- 54.9. TestSubnormallyMonomial
- 54.10. TestMonomialQuick
- 54.11. TestMonomial
- 54.12. TestRelativelySM
- 54.13. IsMinimalNonmonomial
- 54.14. MinimalNonmonomialGroup
55. Getting and Installing GAP
- 55.1. Getting GAP
- 55.2. Upgrading GAP
- 55.3. GAP for UNIX
- 55.4. Installation of GAP for UNIX
- 55.5. Features of GAP for UNIX
- 55.6. GAP for MS-DOS
- 55.7. Copyright of GAP for MS-DOS
- 55.8. Installation of GAP for MS-DOS
- 55.9. Features of GAP for MS-DOS
- 55.10. GAP for TOS
- 55.11. Copyright of GAP for TOS
- 55.12. Installation of GAP for TOS
- 55.13. Features of GAP for TOS
- 55.14. Porting GAP
56. Share Libraries
- 56.1. RequirePackage
- 56.2. ANU pq Package
- 56.3. Installing the ANU pq Package
- 56.4. ANU Sq Package
- 56.5. Installing the ANU Sq Package
- 56.6. GRAPE Package
- 56.7. Installing the GRAPE Package
- 56.8. MeatAxe Package
- 56.9. Installing the MeatAxe Package
- 56.10. NQ Package
- 56.11. Installing the NQ Package
- 56.12. SISYPHOS Package
- 56.13. Installing the SISYPHOS Package
- 56.14. Smash Package
- 56.15. Installing the Smash Package
- 56.16. Vector Enumeration Package
- 56.17. Installing the Vector Enumeration Package
- 56.18. Weyl Package
- 56.19. Installing the Weyl Package
- 56.20. The XGap Package
- 56.21. Installing the XGap Package
57. ANU Pq
- 57.1. Pq
- 57.2. PqHomomorphism
- 57.3. PqDescendants
- 57.4. PqList
- 57.5. SavePqList
- 57.6. StandardPresentation
- 57.7. IsIsomorphicPGroup
58. Grape
- 58.1. Functions to construct and modify graphs
- 58.2. Graph
- 58.3. EdgeOrbitsGraph
- 58.4. NullGraph
- 58.5. CompleteGraph
- 58.6. JohnsonGraph
- 58.7. AddEdgeOrbit
- 58.8. RemoveEdgeOrbit
- 58.9. AssignVertexNames
- 58.10. Functions to inspect graphs, vertices and edges
- 58.11. IsGraph
- 58.12. OrderGraph
- 58.13. IsVertex
- 58.14. VertexName
- 58.15. Vertices
- 58.16. VertexDegree
- 58.17. VertexDegrees
- 58.18. IsLoopy
- 58.19. IsSimpleGraph
- 58.20. Adjacency
- 58.21. IsEdge
- 58.22. DirectedEdges
- 58.23. UndirectedEdges
- 58.24. Distance
- 58.25. Diameter
- 58.26. Girth
- 58.27. IsConnectedGraph
- 58.28. IsBipartite
- 58.29. IsNullGraph
- 58.30. IsCompleteGraph
- 58.31. Functions to determine regularity properties of graphs
- 58.32. IsRegularGraph
- 58.33. LocalParameters
- 58.34. GlobalParameters
- 58.35. IsDistanceRegular
- 58.36. CollapsedAdjacencyMat
- 58.37. OrbitalGraphIntersectionMatrices
- 58.38. Some special vertex subsets of a graph
- 58.39. ConnectedComponent
- 58.40. ConnectedComponents
- 58.41. Bicomponents
- 58.42. DistanceSet
- 58.43. Layers
- 58.44. IndependentSet
- 58.45. Functions to construct new graphs from old
- 58.46. InducedSubgraph
- 58.47. DistanceSetInduced
- 58.48. DistanceGraph
- 58.49. ComplementGraph
- 58.50. PointGraph
- 58.51. EdgeGraph
- 58.52. UnderlyingGraph
- 58.53. QuotientGraph
- 58.54. BipartiteDouble
- 58.55. GeodesicsGraph
- 58.56. CollapsedIndependentOrbitsGraph
- 58.57. CollapsedCompleteOrbitsGraph
- 58.58. NewGroupGraph
- 58.59. Vertex-Colouring and Complete Subgraphs
- 58.60. VertexColouring
- 58.61. CompleteSubgraphs
- 58.62. CompleteSubgraphsOfGivenSize
- 58.63. Functions depending on nauty
- 58.64. AutGroupGraph
- 58.65. IsIsomorphicGraph
- 58.66. An example
59. The MeatAxe
- 59.1. More about the MeatAxe in GAP
- 59.2. GapObject
- 59.3. Using the MeatAxe in GAP. An Example
- 59.4. MeatAxe Matrices
- 59.5. MeatAxeMat
- 59.6. Operations for MeatAxe Matrices
- 59.7. Functions for MeatAxe Matrices
- 59.8. BrauerCharacterValue
- 59.9. MeatAxe Permutations
- 59.10. MeatAxePerm
- 59.11. Operations for MeatAxe Permutations
- 59.12. Functions for MeatAxe Permutations
- 59.13. MeatAxe Matrix Groups
- 59.14. Functions for MeatAxe Matrix Groups
- 59.15. MeatAxe Matrix Algebras
- 59.16. Functions for MeatAxe Matrix Algebras
- 59.17. MeatAxe Modules
- 59.18. Set Theoretic Functions for MeatAxe Modules
- 59.19. Vector Space Functions for MeatAxe Modules
- 59.20. Module Functions for MeatAxe Modules
- 59.21. MeatAxe.Unbind
- 59.22. MeatAxe Object Records
60. Sisyphos
- 60.1. PrintSISYPHOSWord
- 60.2. PrintSisyphosInputPGroup
- 60.3. IsCompatiblePCentralSeries
- 60.4. Automorphisms
- 60.5. AgNormalizedAutomorphisms
- 60.6. AgNormalizedOuterAutomorphisms
- 60.7. IsIsomorphic
- 60.8. Isomorphisms
- 60.9. CorrespondingAutomorphism
- 60.10. AutomorphismGroupElements
- 60.11. NormalizedUnitsGroupRing
61. Smash, Matrix Groups and G-Modules
- 61.1. GModule
- 61.2. IsGModule
- 61.3. DualGMod
- 61.4. InducedGMod
- 61.5. PermGMod
- 61.6. TensorProductGMod
- 61.7. WedgeGMod
- 61.8. WreathProd
- 61.9. WreathPower
- 61.10. IsIrredGMod
- 61.11. SubQuotGMod
- 61.12. SpinBasis
- 61.13. IsAbsIrredGMod
- 61.14. FieldGenCentMat
- 61.15. IsomGMod
- 61.16. HomGMod
- 61.17. MinSubGMods
- 61.18. ChopGMod
- 61.19. SmashGMod
- 61.20. SemiLinearDecomp
- 61.21. TensorProductDecomp
- 61.22. SymTensorProductDecomp
- 61.23. ExtraSpecialDecomp
- 61.24. IsPrimitiveGMod
- 61.25. MinBlocks
- 61.26. BlockSystemFlag
- 61.27. InitialiseSeed
- 61.28. RandomElement
- 61.29. ChooseRandomElements
- 61.30. ElementOfOrder
- 61.31. ElementWithCharPol
- 61.32. LargestPrimeOrderElement
- 61.33. LargestPrimePowerOrderElement
- 61.34. MatrixOrder
- 61.35. Other matrix functions
- 61.36. Components of a G-module record
- 61.37. Examples
62. Vector Enumeration
- 62.1. Operation for Finitely Presented Algebras
- 62.2. More about Vector Enumeration
- 62.3. Examples of Vector Enumeration
- 62.4. Using Vector Enumeration with the MeatAxe
63. Weyl Groups and Hecke Algebras
- 63.1. CartanMat
- 63.2. DirectSumCartanMat
- 63.3. SimpleReflectionMatrices
- 63.4. Rootsystem
- 63.5. PermRepresentationRoots
- 63.6. Weyl
- 63.7. PermWeylWord
- 63.8. WeylWordPerm
- 63.9. WeylLengthPerm
- 63.10. ReducedWeylWord
- 63.11. LongestWeylWord
- 63.12. WeylReflections
- 63.13. WeylRightCosetRepresentatives
- 63.14. WeylCosetPermRepresentation
- 63.15. WeylElements
- 63.16. WeylConjugacyClasses
- 63.17. ParametersCentralizers
- 63.18. Bruhat
- 63.19. KazhdanLusztigPolynomial
- 63.20. KLCoefficient
- 63.21. WeylMueMat
- 63.22. DecomposedLeftCells
- 63.23. LeftCells
- 63.24. LeftCellRepresentation
- 63.25. Hecke
- 63.26. WeylClassPolynomials
- 63.27. HeckeReflectionRepresentation
- 63.28. CheckHeckeDefiningRelations
- 63.29. CharHeckeRepresentation
- 63.30. HeckeCharTable
Index

Written by mS 1994/12/27.