The Construction of a Matrix Group
Construction of the General Linear Group
GeneralLinearGroup(n, R) : RngIntElt, Rng -> GrpMat
Example GrpMat_Create (H21E1)
Construction of a Matrix Group Element
elt< G | L > : GrpMat, List(RngElt) -> GrpMatElt
G ! Q : GrpMat, [ RngElt ] -> GrpMatElt
ElementToSequence(g) : GrpMatElt -> [ RngElt ]
Identity(G) : GrpMat -> GrpMatElt
Example GrpMat_Matrices (H21E2)
Construction of a General Matrix Group
MatrixGroup< n, R | L > : RngIntElt, Rng, List -> GrpMat
Example GrpMat_Constructor (H21E3)
Example GrpMat_GLSylow (H21E4)
Changing Rings
ChangeRing(G, S) : GrpMat, Rng -> GrpMat, Map
ChangeRing(G, S, f) : GrpMat, Rng, Map -> GrpMat, Map
RestrictField(G, S) : GrpMat, FldFin -> GrpMat, Map
ExtendField(G, L) : GrpMat, FldFin -> GrpMat, Map
Coercion between Matrix Structures
R ! g : AlgMat, GrpMatElt -> RngMatElt
G ! r : GrpMat, RngMatElt -> GrpMatElt
M ! g : ModMat, GrpMatElt -> ModMatElt
G ! m : GrpMat, ModMatElt -> GrpMatElt
Basic Invariants of a Matrix Group
Accessing a Group
G . i : GrpMat, RngIntElt -> GrpMatElt
Degree(G) : GrpMat -> RngIntElt
Generators(G) : GrpMat -> { GrpMatElt }
NumberOfGenerators(G) : GrpMat -> RngIntElt
CoefficientRing(G) : GrpMat -> Rng
RSpace(G) : GrpMat -> ModTupRng
VectorSpace(G) : GrpMat -> ModTupFld
GModule(G) : GrpMat -> ModGrp
Generic(G) : GrpMat -> GrpMat
Parent(G) : GrpMatElt -> GrpMat
Group Order
IsFinite(G) : GrpMat -> Bool, RngIntElt
Order(G) : GrpMat -> RngIntElt
FactoredOrder(G) : GrpMat -> [ <RngIntElt, RngIntElt> ]
Example GrpMat_Order (H21E5)
Abstract Group Predicates
IsAbelian(G) : GrpMat -> BoolElt
IsCyclic(G) : GrpMat -> BoolElt
IsElementaryAbelian(G) : GrpMat -> BoolElt
IsNilpotent(G) : GrpMat -> BoolElt
IsSoluble(G) : GrpMat -> BoolElt
IsPerfect(G) : GrpMat -> BoolElt
IsSimple(G) : GrpMat -> BoolElt
Example GrpMat_Order (H21E6)
Homomorphisms
hom<G | L> : GrpMat, List -> Map
Domain(f) : Map -> Grp
Codomain(f) : Map -> Grp
Image(f) : Map -> Grp
Kernel(f) : Map -> Grp
IsHomomorphism(G, H, Q) : GrpMat, GrpMat, SeqEnum[GrpMatElt] -> Bool, Map
Example GrpMat_Homomorphism (H21E7)
Chevalley Groups
ChevalleyGroup(s, n, K: parameters) : MonStgElt, RngIntElt, FldFin -> GrpMat
General and Special Linear Groups
GeneralLinearGroup(arguments)
AffineGeneralLinearGroup(arguments)
SpecialLinearGroup(arguments)
AffineSpecialLinearGroup(arguments)
General and Special Unitary Groups
GeneralUnitaryGroup(arguments)
SpecialUnitaryGroup(arguments)
Symplectic Groups
SymplecticGroup(arguments)
Orthogonal Groups
GeneralOrthogonalGroup(arguments)
GeneralOrthogonalGroupPlus(arguments)
GeneralOrthogonalGroupMinus(arguments)
SpecialOrthogonalGroup(arguments)
SpecialOrthogonalGroupPlus(arguments)
SpecialOrthogonalGroupMinus(arguments)
Omega(arguments)
OmegaPlus(arguments)
OmegaMinus(arguments)
Suzuki Groups
SuzukiGroup(arguments)
Example GrpMat_Symplectic (H21E8)
Example GrpMat_Suzuki (H21E9)
Construction of Extensions
DirectProduct(G, H) : GrpMat, GrpMat -> GrpMat
DirectProduct(Q) : [ GrpMat ] -> GrpMat
SemiLinearGroup(G, S) : GrpMat, FldFin -> GrpMat
TensorWreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
WreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
Example GrpMat_Constructions (H21E10)
Arithmetic with Matrices
g * h : GrpMatElt, GrpMatElt -> GrpMatElt
g ^ n : GrpMatElt, RngIntElt -> GrpMatElt
g / h : GrpMatElt, GrpMatElt -> GrpMatElt
g ^ h : GrpMatElt, GrpMatElt -> GrpMatElt
(g, h) : GrpMatElt, GrpMatElt -> GrpMatElt
(g_1, ..., g_r) : GrpMatElt, ..., GrpMatElt -> GrpMatElt
Example GrpMat_Arithmetic (H21E11)
Predicates for Matrices
g eq h : GrpMatElt, GrpMatElt -> BoolElt
g ne h : GrpMatElt, GrpMatElt -> BoolElt
IsIdentity(g) : GrpMatElt -> BoolElt
IsScalar(g) : GrpMatElt -> BoolElt
Matrix Invariants
Degree(g) : GrpMatElt -> RngIntElt
HasFiniteOrder(g) : GrpMatElt -> BoolElt, RngIntElt
Order(g) : GrpMatElt -> RngIntElt
FactoredOrder(g) : GrpMatElt -> [ <RngIntElt, RngIntElt> ]
Determinant(g) : GrpMatElt -> RngElt
Trace(g) : GrpMatElt -> RngElt
CharacteristicPolynomial(g: parameters) : GrpMatElt -> RngPolElt
MinimalPolynomial(g) : GrpMatElt -> RngPolElt
Example GrpMat_Invariants (H21E12)
Membership and Equality
g in G : GrpMatElt, GrpMat -> BoolElt
g notin G : GrpMatElt, GrpMat -> BoolElt
S subset G : { GrpMatElt }, GrpMat -> BoolElt
H subset G : GrpMat, GrpMat -> BoolElt
S notsubset G : { GrpMatElt }, GrpMat -> BoolElt
H notsubset G : GrpMat, GrpMat -> BoolElt
H eq G : GrpMat, GrpMat -> BoolElt
H ne G : GrpMat, GrpMat -> BoolElt
Set Operations
NumberingMap(G) : GrpMat -> Map
RandomProcess(G) : GrpMat -> Process
Random(G: parameters) : GrpMat -> GrpMatElt
Random(P) : Process -> GrpMatElt
Example GrpMat_Random (H21E13)
Conjugacy
Class(H, x) : GrpMat, GrpMatElt -> { GrpMatElt }
ClassMap(G) : GrpMat -> Map
ConjugacyClasses(G: parameters) : GrpMat -> [ < RngIntElt, RngIntElt, GrpMatElt > ]
ClassRepresentative(G, x) : GrpMat, GrpMatElt -> GrpMatElt
IsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt | Unass
[Future release] IsConjugate(G, H, K) : GrpMat, GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
Exponent(G) : GrpMat -> RngIntElt
NumberOfClasses(G) : GrpMat -> RngIntElt
PowerMap(G) : GrpMat -> Map
AssertAttribute(G, "Classes", Q) : GrpMat, MonStgElt, [ GrpMatElt ] ->
Example GrpMat_RationalMatrixGroupDatabase (H21E14)
Construction of Subgroups
sub<G | L> : GrpMat, List -> GrpMat
ncl<G | L> : GrpMat, List -> GrpMat
Example GrpMat_Subgroups (H21E15)
Elementary Properties of Subgroups
Index(G, H) : GrpMat, GrpMat -> RngIntElt
FactoredIndex(G, H) : GrpMat, GrpMat -> [ <RngIntElt, RngIntElt> ]
IsCentral(G, H) : GrpMat -> BoolElt
IsMaximal(G, H) : GrpMat, GrpMat -> BoolElt
IsNormal(G, H) : GrpMat, GrpMat -> BoolElt
IsSubnormal(G, H) : GrpMat, GrpMat -> BoolElt
Standard Subgroups
H ^ g : GrpMat, GrpMatElt -> GrpMat
H meet K : GrpMat, GrpMat -> GrpMat
CommutatorSubgroup(G, H, K) : GrpMat, GrpMat, GrpMat -> GrpMat
Centralizer(G, g) : GrpMat, GrpMatElt -> GrpMat
Centralizer(G, H) : GrpMat, GrpMat -> GrpMat
Core(G, H) : GrpMat, GrpMat -> GrpMat
H ^ G : GrpMat, GrpMat -> GrpMat
[Future release] Normalizer(G, H) : GrpMat, GrpMat -> GrpMat
SylowSubgroup(G, p) : GrpMat, RngIntElt -> GrpMat
pCore(G, p) : GrpMat, RngIntElt -> GrpMat
Low Index Subgroups
LowIndexSubgroups(G, R : parameters) : GrpMat, RngIntElt -> [ GrpMat ]
Example GrpMat_LowIndexMatrixGroup (H21E16)
Construction of Quotient Groups
quo<G | L> : GrpMat, List -> GrpPerm, Map
G / N : GrpMat, GrpMat -> GrpMat
Example GrpMat_Quotient (H21E17)
Abelian, Nilpotent and Soluble Quotients
AbelianQuotient(G) : GrpMat -> GrpAb, Map
ElementaryAbelianQuotient(G, p) : GrpMat, RngIntElt -> GrpAb, Map
pQuotient(G, p, c) : GrpMat, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt
NilpotentQuotient(G, c) : GrpMat, RngIntElt -> GrpGPC, Map
SolvableQuotient(G): GrpMat -> GrpPC, Map
PCGroup(G): GrpMat -> GrpPC, Map
Example GrpMat_SpecialQuotient (H21E18)
Orbits and Stabilizers
u * g : ModTupElt, GrpMatElt -> ModTupElt
y ^ g : Elt, GrpMatElt -> Elt
y ^ G : Elt, GrpMat -> SetEnum
OrbitBounded(G, y, b) : GrpMat, Elt, RngIntElt -> BoolElt, SetEnum
Orbits(G) : GrpMat -> [ SetIndx ]
LineOrbits(G) : GrpMat -> [ SetIndx ]
OrbitClosure(G, S) : GrpMat, { Elt } -> GSet
Stabilizer(G, y) : GrpMat, Elt -> GrpMat
Example GrpMat_Orbits (H21E19)
Orbit and Stabilizer Functions for Large Groups
OrbitsOfSpaces(G, k) : GrpMat, RngIntElt -> SeqEnum
NumberOfFixedSpaces(x, s) : GrpMatElt, RngIntElt -> RngIntElt
Example GrpMat_OrbitsOfSpaces (H21E20)
EstimateOrbit(G, U: parameters) : GrpMat, ModTupFld -> RngIntElt, RngIntElt, RngIntElt
ApproximateStabiliser(G, A, U: parameters) : GrpMat, GrpMat, ModTupFld -> GrpMat, GrpMat, RngIntElt, RngIntElt, RngIntElt
Example GrpMat_OrbitsOfSpaces (H21E21)
StabiliserOfSpaces(Q) : Spaces -> GrpMat, SeqEnum
Example GrpMat_StabiliserOfSpaces (H21E22)
UnipotentStabiliser(G, U: parameters) : Grp, ModTupFld -> GrpMat, ModTupFld, GrpMatElt
Example GrpMat_UnipotentStabiliser (H21E23)
Action on Orbits
OrbitAction(G, T) : GrpMat, Elt -> Hom(Grp), GrpPerm, GrpMat
OrbitActionBounded(G, T, b) : GrpMat, Elt, RngIntElt -> BoolElt, Hom(Grp), GrpPerm, GrpMat
OrbitImage(G, T) : GrpMat, Set -> GrpPerm
OrbitImageBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpPerm
OrbitKernel(G, T) : GrpMat, Set -> GrpMat
OrbitKernelBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpMat
Example GrpMat_Actions (H21E24)
Action on a Coset Space
CosetAction(G, H) : GrpMat, GrpMat -> Hom(Grp), GrpPerm, GrpMat
CosetImage(G, H) : GrpMat, GrpMat -> GrpPerm
CosetKernel(G, H) : GrpMat, GrpMat -> GrpMat
Example GrpMat_CosetAction (H21E25)
Action on the Natural G-Module
GModule(G) : GrpMat -> ModGrp
IsIrreducible(G) : GrpMat -> BoolElt, ModGrp
SubmoduleAction(G, S) : GrpMat -> Map, GrpMat
SubmoduleImage(G, S) : GrpMat -> GrpMat
QuotientModuleAction(G, S) : GrpMat -> Map, GrpMat
QuotientModuleImage(G, S) : GrpMat -> GrpMat
IsAbsolutelyIrreducible(G) : GrpMat -> BoolElt
AbsoluteRepresentation(M) : GrpMat -> GrpMat
MinimalField(G) : GrpMat -> FldFin
Normal and Subnormal Subgroups
Characteristic Subgroups and Subgroup Series
Centre(G) : GrpMat -> GrpMat
DerivedLength(G) : GrpMat -> RngIntElt
DerivedSeries(G) : GrpMat -> [ GrpMat ]
DerivedSubgroup(G) : GrpMat -> GrpMat
[Future release] FittingSubgroup(G) : GrpMat -> GrpMat
LowerCentralSeries(G) : GrpMat -> [ GrpMat ]
NilpotencyClass(G) : GrpMat -> RngIntElt
H ^ G : GrpMat -> GrpMat
SolubleResidual(G) : GrpMat -> GrpMat
SubnormalSeries(G, H) : GrpMat, GrpMat -> [ GrpMat ]
UpperCentralSeries(G) : GrpMat -> [ GrpMat ]
Example GrpMat_Series (H21E26)
The Soluble Radical and its Quotient
Radical(G) : GrpMat -> GrpMat
RadicalQuotient(G) : GrpMat -> GrpPerm, Hom(Grp), GrpMat
ElementaryAbelianSeries(G: parameters) : GrpPerm -> [ GrpPerm ]
Composition and Chief Factors
CompositionFactors(G) : : GrpMat -> [ <RngIntElt, RngIntElt, RngIntElt> ]
ChiefFactors(G) : GrpMat -> [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
ChiefSeries(G) : GrpMat -> [ GrpMat ], [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
Example GrpMat_CompositionFactors (H21E27)
Coset Tables and Transversals
CosetTable(G, H) : Grp, Grp -> Hom(Grp)
Transversal(G, H) : GrpMat, GrpMat -> {@ GrpMatElt @}, Map
Generators and Relations
FPGroup(G) : GrpMat :-> GrpFP, Hom(Grp)
FPGroupStrong(G) : GrpMat :-> GrpFP, Hom(Grp)
Matrices as Words
WordGroup(G) : GrpMat -> GrpSLP, Map
InverseWordMap(G) : GrpMat -> Map
Representation Theory
LinearCharacters(G) : GrpMat -> [ Chtr ]
CharacterTable(G) : GrpMat -> TabChtr
PermutationCharacter(G, H) : GrpMat, GrpMat -> AlgChtrElt
GModule(G) : GrpMat -> ModGrp
GModule(G, A) : GrpMat, AlgMat -> ModGrp
GModule(G, Q) : GrpMat, [ AlgMatElt ] -> ModGrp
GModule(G, A, B) : GrpMat, GrpMat, GrpMat -> ModGrp, Map
PermutationModule(G, H, R) : GrpMat, GrpMat, Rng -> ModGrp
Example GrpMat_GModule (H21E28)
Base and Strong Generating Set
Controlling Selection of a Base
GoodBasePoints(G: parameters) : GrpMat -> []
AssertAttribute(G, "Base", B) : GrpMat, MonStgElt, Tup ->
HasAttribute(G, "Base") : GrpMat, MonStgElt -> BoolElt, Tup
AssertAttribute(GrpMat, "FirstBasicOrbitBound", n) : Cat, MonStgElt, RngIntElt ->
HasAttribute(GrpMat, "FirstBasicOrbitBound") : Cat, MonStgElt -> BoolElt, RngIntElt
Construction of a Base and Strong Generating Set
BSGS(G) : GrpMat ->
RandomSchreier(G: parameters) : GrpMat ->
ToddCoxeterSchreier(G) : GrpMat : ->
Verify(G) : GrpMat ->
Defining Values for Attributes
AssertAttribute(G, "Order", n) : GrpMat, MonStgElt, RngIntElt ->
AssertAttribute(G, "IsVerified", b) : GrpMat, MonStgElt, BoolElt ->
HasAttribute(G, "Order") : GrpMat, MonStgElt -> RngIntElt
HasAttribute(G, "IsVerified") : GrpMat, MonStgElt -> BoolElt
Accessing the Base and Strong Generating Set
Base(G) : GrpMat -> [Elt]
BasePoint(G, i) : GrpMat, RngIntElt -> Elt
BasicOrbit(G, i) : GrpMat, RngIntElt -> SetIndx
BasicOrbitLength(G, i) : GrpMat, RngIntElt -> RngIntElt
BasicOrbitLengths(G) : GrpMat -> [RngIntElt]
BasicStabilizer(G, i) : GrpMat, RngIntElt -> GrpMat
BasicStabilizerChain(G) : GrpMat -> [GrpMat]
NumberOfStrongGenerators(G) : GrpMat -> RngIntElt
StrongGenerators(G) : GrpMat -> SetIndx(GrpMat)
Conversion to a PC-Group
PolycyclicGenerators(G) : GrpMat -> [ GrpPCElt ]
PCGroup(G) : GrpMat -> GrpPC, Map
Soluble Group Functions
pCentralSeries(G, p) : GrpMat, RngIntElt -> [ GrpMat ]
p-group Functions
IsSpecial(G) : GrpMat -> BoolElt
IsExtraSpecial(G) : GrpMat -> BoolElt
FrattiniSubgroup(G) : GrpMat -> GrpMat
JenningsSeries(G) : GrpMat -> [ GrpMat ]
Abelian Group Functions
AbelianInvariants(G) : GrpMat -> [ RngIntElt ]
Classical forms
ClassicalForms(G): GrpMat -> BoolElt
SymplecticForm(G) : GrpMat -> AlgMatElt
ScalarsSymplecticForm(G) : GrpMat -> SeqEnum
SymmetricBilinearForm(G) : GrpMat -> AlgMatElt
ScalarsSymmetricBilinearForm(G) : GrpMat -> SeqEnum
QuadraticForm(G): GrpMat -> AlgMatElt
ScalarsQuadraticForm(G) : GrpMat -> SeqEnum
UnitaryForm(G) : GrpMat -> AlgMatElt
ScalarsUnitaryForm(G) : GrpMat -> SeqEnum
FormType(G) : GrpMat -> MonStgElt
Example GrpMat_ClassicalForms (H21E29)
Recognizing Classical Groups in their Natural Representation
RecognizeClassical( G : parameters): GrpMat -> BoolElt
IsLinearGroup(G) : GrpMat -> BoolElt
IsSymplecticGroup(G) : GrpMat -> BoolElt
IsOrthogonalGroup(G) : GrpMat ->BoolElt
IsUnitaryGroup(G) : GrpMat -> BoolElt
ClassicalType(G) : GrpMat -> MonStgElt
Example GrpMat_RecognizeClassical (H21E30)
Primitivity Testing
IsPrimitive(G: parameters) : GrpMat -> BoolElt
BlockSystem(G) : GrpMat -> Rec
BlocksImage(G) : GrpMat -> GrpPerm
Example GrpMat_IsPrimitive (H21E31)
Semilinearity
IsSemiLinear(G) : GrpMat -> BoolElt
DegreeOfFieldExtension(G) : GrpMat -> RngIntElt
CentralisingMatrix(G) : GrpMat -> AlgMatElt
FrobeniusAutomorphisms(G) : GrpMat -> SeqEnum
WriteOverLargerField(G) : GrpMat -> GrpMat, GrpMat, GrpAb, SeqEnum
Example GrpMat_Semilinearity (H21E32)
Tensor Products
IsTensor(G: parameters) : GrpMat -> BoolElt
TensorBasis(G) : GrpMat -> GrpMatElt
TensorFactors(G) : GrpMat -> GrpMat, GrpMat
IsProportional(X, k) : Mtrx, RngIntElt -> BoolElt, Tup
Example GrpMat_Tensor (H21E33)
Tensor-induced Groups
IsTensorInduced(G : parameters) : GrpMat -> BoolElt
TensorInducedBasis(G) : GrpMat -> GrpMatElt
TensorInducedPermutations(G) : GrpMat -> SeqEnum
Example GrpMat_TensorInduced (H21E34)
Decompositions with Respect to a Normal Subgroup
SearchForDecomposition(G, S) : GrpMat, [GrpMatElt] -> BoolElt
Accessing the Decomposition Information
Example GrpMat_Decompose (H21E35)
IsExtraSpecialNormalise(G) : GrpMat -> BoolElt
ExtraSpecialParameters(G) : GrpMat -> [RngIntElt, RngIntElt]
ExtraSpecialGroup(G) : GrpMat -> GrpMat
Example GrpMat_ExtraSpecial (H21E36)
Writing Representations over Subfields
IsOverSmallerField(G) : GrpMat -> BoolElt, GrpMat
IsOverSmallerField(G, d) : GrpMat, RngIntElt -> BoolElt, GrpMat
Example GrpMat_IsOverSmallerField (H21E37)
Related Functions
WriteOverSmallerField(G, F) : GrpMat, FldFin -> GrpMat, Map