- Introduction
- Creation of a Matrix Group
- Construction of the General Linear Group
- Construction of a Matrix Group Element
- Construction of a General Matrix Group
- 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
- Accessing Associated Structures
- Homomorphisms
- Operations on Matrices
- 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
- (g1, ..., gr) : GrpMatElt, ..., GrpMatElt -> GrpMatElt
- Example GrpMatGen_Arithmetic (H65E7)
- Predicates for Matrices
- Matrix Invariants
- Global Properties
- Abstract Group Predicates
- Conjugacy
- Class(H, x) : GrpMat, GrpMatElt -> { GrpMatElt }
- ClassMap(G) : GrpMat -> Map
- ConjugacyClasses(G: parameters) : GrpMat -> [ < RngIntElt, RngIntElt, GrpMatElt > ]
- ClassRepresentative(G, x) : GrpMat, GrpMatElt -> GrpMatElt
- ClassCentraliser(G, i) : GrpMat, RngIntElt -> GrpMat
- ClassRepresentativeFromInvariants(G, p, h, t) : GrpMat, SeqEnum, SeqEnum, FldFinElt -> GrpMatElt
- IsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt | Unass
- NumberOfClasses(G) : GrpMat -> RngIntElt
- PowerMap(G) : GrpMat -> Map
- AssertAttribute(G, "Classes", Q) : GrpMat, MonStgElt, SeqEnum ->
- Example GrpMatGen_RationalMatrixGroupDatabase (H65E12)
- Conjugacy in Classical Groups
- DualPolynomial(f) : RngUPolElt -> RngUPolElt
- StarIrreduciblePolynomials(F,d) : FldFin, RngIntElt -> SeqEnum
- PhiDual(f,phi) : RngUPolElt -> RngUPolElt
- PhiIrreduciblePolynomials(F,d) : FldFin, RngIntElt -> SeqEnum[Tup]
- ExtendedSymplecticGroup(n,q,m) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
- IndexOfSp(G) : GrpMat -> RngIntElt
- TildeDualPolynomial(f) : RngUPolElt -> RngUPolElt
- TildeIrreduciblePolynomials(q,d) : RngIntElt, RngIntElt -> SeqEnum
- ExtendedSpecialUnitaryGroup(n,q,m) : RngIntElt, RngIntElt, RngIntElt -> GrpMat
- IndexOfSU(G) : GrpMat -> RngIntElt
- ClassicalConjugacyClasses(G) : GrpMat -> SeqEnum, SetIndx
- ClassicalConjugacyClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, SetIndx
- ClassicalCentralizer(G,g) : GrpMat, GrpMatElt -> GrpMat
- ClassicalCentraliserOrder(G,g) : GrpMat, GrpMatElt -> RngIntEltFact
- ClassicalClassSize(G,g) : GrpMat, GrpMatElt -> RngIntElt
- ClassicalIsConjugate(G,g,h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt
- Example GrpMatGen_Class-calculations-I (H65E13)
- ClassicalClassMap(G) : GrpMat -> Map
- ClassesForFixedSemisimple(G,x) : GrpMat, GrpMatElt -> SeqEnum, SetIndx
- IsometryGroupClassLabel(type, g) : MonStgElt, GrpMatElt -> SetMulti
- Example GrpMatGen_Class-calculations (H65E14)
- Example GrpMatGen_Class-calculations-III (H65E15)
- Example GrpMatGen_Invlayer (H65E16)
- UnipotentClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, SeqEnum
- SemisimpleClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum
- IsometryGroupNumberOfClasses(type, n): MonStgElt, RngIntElt -> RngUPolElt
- Example GrpMatGen_Class-calculations-IV (H65E17)
- ProjectiveClassicalClasses(type,d,q) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum, GrpPerm, HomGrp, SeqEnum
- Example GrpMatGen_ProjectiveClasses (H65E18)
- Example GrpMatGen_ProjWithMatrices (H65E19)
- Subgroups
- Construction of Subgroups
- Elementary Properties of Subgroups
- Index(G, H) : GrpMat, GrpMat -> RngIntElt
- FactoredIndex(G, H) : GrpMat, GrpMat -> [ <RngIntElt, RngIntElt> ]
- IsCentral(G, H) : GrpMat, 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
- Centraliser(G, g) : GrpMat, GrpMatElt -> GrpMat
- Centraliser(G, H) : GrpMat, GrpMat -> GrpMat
- Core(G, H) : GrpMat, GrpMat -> GrpMat
- H ^ G : GrpMat, GrpMat -> GrpMat
- Normalizer(G, H) : GrpMat, GrpMat -> GrpMat
- GLNormalizer(H : parameter) : GrpMat -> GrpMat
- SylowSubgroup(G, p) : GrpMat, RngIntElt -> GrpMat
- pCore(G, p) : GrpMat, RngIntElt -> GrpMat
- Low Index Subgroups
- Conjugacy Classes of Subgroups
- SubgroupClasses(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
- MaximalSubgroups(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
- MaximalSubgroups(G,N: parameters) : GrpMat, GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
- SubgroupsLift(G, A, B, Q: parameters) : GrpMat, GrpMat, GrpMat, SeqEnum -> SeqEnum
- IsConjugate(G, H, K) : GrpMat, GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
- IsGLConjugate(H, K) : GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
- Quotient Groups
- Construction of Quotient Groups
- 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 GrpMatGen_SpecialQuotient (H65E23)
- Matrix Group Actions
- Orbits and Stabilizers
- u * g : ModTupRngElt, GrpMatElt -> ModTupRngElt
- 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 GrpMatGen_Orbits (H65E24)
- Orbit and Stabilizer Functions for Large Groups
- OrbitsOfSpaces(G, k) : GrpMat, RngIntElt -> SeqEnum
- NumberOfFixedSpaces(x, s) : GrpMatElt, RngIntElt -> RngIntElt
- Example GrpMatGen_OrbitsOfSpaces (H65E25)
- EstimateOrbit(G, v: parameters) : GrpMat, ModTupFldElt -> RngIntElt, RngIntElt, RngIntElt
- ApproximateStabiliser(G, A, U: parameters) : GrpMat, GrpMat, ModTupFld -> GrpMat, GrpMat, RngIntElt, RngIntElt, RngIntElt
- Example GrpMatGen_OrbitsOfSpaces (H65E26)
- StabiliserOfSpaces(Q) : SeqEnum -> GrpMat, SeqEnum
- Example GrpMatGen_StabiliserOfSpaces (H65E27)
- IsUnipotent(G) : GrpMat -> BoolElt
- UnipotentStabiliser(G, U: parameters) : GrpMat, ModTupFld -> GrpMat, ModTupFld, GrpMatElt, GrpSLPElt
- Example GrpMatGen_UnipotentStabiliser (H65E28)
- 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, SetIndx
- OrbitImageBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpPerm, SetIndx
- OrbitKernel(G, T) : GrpMat, Set -> GrpMat
- OrbitKernelBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpMat
- Example GrpMatGen_Actions (H65E29)
- Action on a Coset Space
- Action on the Natural G-Module
- Normal and Subnormal Subgroups
- Coset Tables and Transversals
- Presentations
- Automorphism Groups
- Representation Theory
- LinearCharacters(G) : GrpMat -> [ Chtr ]
- CharacterTable(G: parameters) : 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
- ChangeOfBasisMatrix(G, S) : GrpMat, ModGrp -> AlgMatElt
- Example GrpMatGen_GModule (H65E35)
- Base and Strong Generating Set
- Introduction
- 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
- 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
- Soluble Matrix Groups
- Bibliography
V2.28, 13 July 2023