GAP
Summary
GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more.
GAP is developed by international cooperation. The system, including source, is distributed freely under the terms of the GNU General Public License. You can study and easily modify or extend GAP for your special use.
The current version is GAP 4, the older version GAP 3 is still available.
Authors
The GAP Group, see the GAP homepage. For each person involved with GAP you can use the search facility for GAP (http://www.gap-system.org/search.html) to find all places in the GAP documentation (website, manuals, Forum archive) in which the name of the person appears.
Vendor
GAP is a free system and distributed without charge by The GAP Group.
Status
officially approved by the authors
Aims and scope
Mathematical Classification
- 01 Discrete Mathematics
- 01.01 Convex and discrete geometry
- 01.02 Graph theory
- 01.03 Enumerative combinatorics
- 01.04 Algebraic combinatorics
- 01.06 Ordered structures
- 02 Algebra / Number Theory
- 02.02 Number theory
- 02.03 Field theory
- 02.04 Commutative rings and algebras
- 02.04.01 Ideals, modules, homomorphisms
- 02.04.02 Polynomial and power series rings
- 02.04.03 Special rings
- 02.04.07 Factorization and primary decomposition
- 02.04.08 Syzygies and resolutions
- 02.04.10 Groebner bases
- 02.05 Linear and multilinear algebra; matrix theory
- 02.05.01 Linear equations
- 02.05.02 Eigenvalues, singular values, and eigenvectors
- 02.05.03 Canonical forms
- 02.05.04 Matrix factorization
- 02.05.05 Integral matrices
- 02.05.06 Multilinear algebra
- 02.06 Non-commutative and general rings and algebras
- 02.06.01 Lie algebras
- 02.07 Category theory; homological algebra
- 02.08 Group theory and generalizations
- 02.08.01 Permutation groups
- 02.08.02 Matrix groups
- 02.08.03 Finitely presented groups
- 02.08.04 Polycyclicly presented groups
- 02.08.05 Black box groups
- 02.08.06 Group actions
- 02.08.07 Subgroup lattices
- 02.08.08 Group cohomology
- 02.08.09 Semigroups
- 02.09 Representation theory
- 02.09.01 Ordinary Representations of Groups
- 02.09.02 Ordinary Representations of Algebras
- 02.09.03 Modular Representations of Groups
- 02.09.04 Modular Representations of Algebras
- 02.09.05 Character Theory
- 02.09.06 Permutation Representations
- 03.01.08 Algebraic groups and geometric invariant theory
- 07 Applications of Mathematics
- 07.87 Theoretical Chemistry
- 07.88 Crystallography
- 08 Teaching
- 08.97 Mathematics education
Keywords
- 1-cohomology
- 2-cohomology
- abelian invariants
- abelian number fields
- absolutely irreducible matrix groups
- actions of matrix groups
- actions of monoids
- actors
- additive magmas
- adjoint representation
- affine crystallographic groups
- ag group
- algebraic extensions of fields
- algebraic number theory
- algebras
- almost Bieberbach group
- almost crystallographic group
- alternating groups
- arithmetic for lists
- arrangements
- associative words
- atlas irrationalities
- atlas of finite groups
- atlas of group representations
- augmented coset tables
- automatic groups
- automorphism groups
- automorphism groups of a graph
- automorphism groups of code
- automorphism groups of design
- automorphism groups of soluble group
- automorphisms and equivalence of character tables
- backtrack
- balanced incomplete-block design
- bases of vector spaces
- basic coset enumeration
- basic groups
- bibd
- binary relations
- block design
- block matrices
- blocks
- block systems
- bol loops
- boolean lists
- booleans
- braid groups
- braid relations
- Brauer characters
- Brauer tables
- Bravais flocks
- Bravais groups
- Bravais lattices
- Bruhat Chevalley order
- burnside groups
- burnside matrices
- Carter subgroups
- cat-1 groups
- center
- central extension
- centralizers
- central series
- cfrac
- character degrees
- character tables
- check matrices
- Chevalley groups
- class functions
- class fusions between character tables
- classical groups
- cliques
- closures of groups
- closures of subgroups
- code
- codeword
- cohomologies
- cohomology groups
- coincidences
- collections
- color groups
- combinations
- combinatorial numbers
- combinatorics
- commutative algebra
- commutators
- commutator subgroups
- complements
- complete subgraphs
- complex reflection groups
- composition series of triangularizable groups
- computer algebra
- congruences for semigroups
- conjugacy classes
- conjugacy classes in classical groups
- conjugacy classes in solvable groups
- conjugacy of matrix groups
- conjugacy problem
- consistency
- consistency check
- constructing character tables
- constructing groups of small order
- constructive representation theory of finite groups
- constructors for basic groups
- continued fraction algorithm
- continued fractions
- conway polynomials
- copyright notice
- coset enumeration
- cosets
- coset tables
- coset table standardization
- covering groups
- covering radius
- covering subgroups
- coxeter groups
- creating finitely presented semigroups
- crossed modules
- crystal graphs
- crystallographic groups
- crystal systems
- Cunningham tables
- cycle structure
- cyclic code
- cyclic split extension method
- cyclotomic numbers
- cyclotomic polynomials
- data bases
- data library
- data library of almost bieberbach groups
- d-class
- decoding
- decomposition matrices
- decomposition numbers
- decomposition of modules
- decompositions
- deductions
- definition of cosets
- derivation
- derived lengths
- derived series
- derived subgroups
- design classification
- design construction
- design isomorphism
- design partitioning
- determinants
- diameters
- digraphs
- directed graphs
- direct products
- direct sum decompositions
- distance-regular graphs
- distance-transitive graphs
- division rings
- Dixon-Schneider algorithm
- Dixons method
- domains
- domains and their elements
- double cosets
- Dynkin diagrams
- echelonized matrices
- ecm
- eigenvalues
- eigenvectors
- elementary divisors
- elements of free magma rings
- elements of given norm
- elliptic curves method
- embeddings and projections for group products
- endomorphisms
- Engel law
- enumerators
- equivalence classes
- equivalence relations
- error-correcting block codes
- euclidean rings
- eulerian function
- exponent law
- expressing group elements as words in generators
- expression trees
- extensions
- exterior powers
- factor groups
- factorizations
- fast signal transforms
- felsch strategy
- fibonacci sequences
- fields
- fields and division rings
- finding submodules
- finite binary relations
- finite fields
- finite geometry
- finite groups
- finitely generated nilpotent groups
- finitely presented algebras
- finitely presented groups
- finitely presented Lie algebras
- finitely-presented modules
- finitely presented monoids
- finitely presented semigroups
- finitely presented semigroups and monoids
- finite perfect groups
- finite presentation of the automorphism group of a free group
- finite presentations
- finite simple groups
- finite state automata
- finite unimodular groups
- fitting classes
- fitting sets
- fitting subgroups
- fixed point free automorphism
- f-normalizer
- fock spaces
- formation
- fourdimensional crystallographic groups
- fourdimensional space groups
- fp group
- Frattini extension method
- Frattini subgroup
- free groups
- free magma rings
- free magmas
- free modules
- free semigroups
- Frobenius automorphism
- Frobenius group
- function fields
- fusions
- Galois action
- Galois conjugacy of cyclotomics
- Galois groups
- Galois groups of Abelian number fields
- gaussian algorithm
- gaussians
- gcd
- generator matrices
- generator polynomials
- generic character tables
- generic degrees
- g-graph
- girth
- gl
- gl(nz)
- Golay codes
- graphical user interface
- graph isomorphism
- graphs
- green class
- green function
- greens relations
- Groebner bases
- group actions
- group automorphisms
- group classes
- group elements
- group homomorphisms
- group libraries
- group products
- group properties
- group representations
- group rings
- groups
- groups of Lie type
- groups of order at most 2000
- groups of order p^6
- groups of order pqr
- groups of small order
- Hall subgroups
- hamming
- Haselgrove Leech Trotter strategy
- h-class
- Hecke algebras
- highest weight module
- Hirsch length
- hlt strategy
- hnn extensions
- homomorphisms of algebras
- Hopf algebra
- hypercenter
- ideal of semigroups
- ideals
- ideals in rings
- ideals of semigroups
- idempotent
- identical relations
- identification of groups
- images
- indeterminates
- induced actions
- induced subgraphs
- infinite groups
- infinity
- injector
- integer factorization
- integers
- integral bases of Abelian number fields
- integral matrices
- integral matrices and lattices
- interactive coset enumeration
- interactive todd coxeter method
- interface to kant
- interface to singular
- international tables for cystallography
- intersections
- intertwining spaces
- invariant forms
- inverse finite automata
- irreducibility tests
- irreducible characters
- irreducible groups
- irreducible maximal finite integral matrix groups
- irreducible solvable matrix groups
- isoclinic groups
- isomorphism testing
- iterators
- Iwahori Hecke algebras
- Kazhdan Lusztig bases
- Kazhdan Lusztig polynomials
- Knuth-Bendix
- Knuth-Bendix procedure
- large degree permutation groups
- lattice of subgroups
- lattice reductions
- lattices
- Laurent polynomials
- l-class
- lcm
- left cell representations
- lexicodes
- libraries
- Lie algebras
- Lie objects
- linear codes
- linear equations
- linear groups
- linear mappings
- list decoding
- lists
- loops
- lower central series
- lower p-central series
- low index subgroups
- lucas sequences
- magma homomorphisms
- magma rings
- magmas
- mappings
- matrices
- matrix
- matrix group recognition
- matrix groups
- matrix representations
- matrix spaces
- maximal order
- maximal subgroups
- meataxe
- metabelian law
- metacyclic groups
- minimal gaps strategy
- minimal nonmonomial groups
- minimal normal subgropus
- minimal polynomials
- minimum distance
- modular group algebra
- module constructions
- module homomorphisms
- modules
- Moebius function
- Molien series
- monoid homomorphism
- monoids
- monomial characters
- monomial orderings
- monomorphisms
- Moufang loop
- mpqs
- multiple polynomial quadratic sieve
- multiplicative arithmetic functions
- multipliers
- multivariate polynomials
- mutable bases
- near fields
- near linear spaces
- near rings
- nilpotent-by-finite groups
- nilpotent Engel groups
- nilpotent Lie algebras
- nilpotent presentations
- nilpotent quotient algorithm
- nonlinear codes
- Nordstrom-Robinson code
- normal forms
- normalizers
- normal structure
- normal subgroups
- nucleus
- number fields
- number theory
- one-point ag codes
- orbits
- orbit-stabilizer algorithms
- orbit stabilizer methods for polycyclic groups
- orderings
- ordinary character tables
- orthogonal embeddings
- p-adic numbers
- parabolic subgroups
- parallel class
- parallel computations
- partial linear spaces
- partially ordered sets
- pc groups
- pcgs
- p-class
- p-cover
- permanent of a matrix
- permutation characters
- permutation groups
- permutation image of an action
- permutations
- p-group generation algorithm
- p-groups
- p-multiplicator
- point group
- Pollards p-1
- polycyclically presented groups
- polycyclic generating sequences
- polycyclic generating systems
- polycyclic groups
- polycyclic presentations
- polycyclic quotient algorithm
- polycylic presentations
- polynomial factorization
- polynomial functions
- polynomial rings
- polynomials
- polynomials and rational functions
- poset
- power maps
- p-quotient algorithm
- preimages
- preimages in the free group
- presentations and tietze transformations
- prime residues
- primitive groups
- primitive permutation groups
- primitive roots and discrete logarithms
- primitivity of characters
- projective characters
- projectors
- q-classes
- quantized enveloping algebra
- quantum fields
- quantum groups
- quasigroups
- quotient methods
- quotients
- radicals
- radical series
- random elements
- random matrices
- ranges
- rational expressions
- rational functions
- rational numbers
- r-class
- recognition of matrix groups
- reconstruction from defining relations
- record assignment
- records
- Reed-Solomon codes
- Rees matrix semigroups
- regular semigroups
- relations
- relators in a presentation
- representations of algebras
- representation theory
- residuals
- residual subgroup
- residue classes
- residue class rings
- resolution
- rewriting
- rewriting in groups and monoids
- rewriting systems
- Richard p Brents factor tables
- right and left Engel elements
- ring homomorphisms
- rings
- root data
- roots modulo integers
- root systems
- row spaces
- row vectors
- Schunck class
- schur cover
- schur multiplicator
- schur multiplier
- Schutzenberger group
- section
- semidirect products
- semigroups
- semisimple Lie algebras
- series of ideals
- sets
- shortcut method
- sign
- sims strategy
- sl
- small groups
- smash meataxe
- socle
- soluble groups
- solvable groups
- solvable word problem
- sorted character tables
- sorted lists
- space groups
- special generating sets
- special multiplication algorithms for matrices over gf(2)
- special pcgs
- stabilizer chains
- stabilizers
- standard generators
- standard generators of groups
- standardization of coset tables
- standard presentation algorithm
- Steiner loop
- straight line programs
- strings and characters
- structured matrices
- subalgebras
- subdesign
- subdirect products
- subfields
- subgroup lattices
- subgroup presentations
- subgroups
- subgroup series
- subgroups of finite index
- submodules
- subnormal subgroups
- Sylow groups
- Sylow subgroups
- symbolic manipulation of group representations
- symmetric groups
- symmetric powers
- symmetrizations of class functions
- symmetry-based matrix factorization
- system normalizer
- table automorphism
- tableaux
- tables of marks
- tails
- t-design
- tensor products
- test solvability
- Tietze transformations
- Todd Coxeter coset enumeration
- Todd Coxeter procedure
- top-c
- torsion subgroup and finite subgroups
- trace fields
- transformation monoids
- transformations
- transformation semigroups
- transitive permutation groups
- transversals
- triality
- triangularizable subgroups
- triangular matrices
- tuples
- unipot
- unipotent elements
- unipotent radicals
- unipotent subgroups
- unit groups
- units
- univariate polynomial rings
- univariate polynomials
- univariate rational functions
- universal enveloping algebras
- unknowns
- upper central series
- upwards extension method
- vector enumeration
- vector space bases
- vector space homomorphisms
- vector spaces
- virtual characters
- virtually nilpotent groups
- visualization
- weight distribution
- weighted power commutator presentation
- Weyl groups
- Williams p 1
- words
- wreath products
- Wyckoff positions
- z-classes