Matrix Groups over Finite Fields
- Introduction
- Finding Elements with Prescribed Properties
- Monte Carlo Algorithms for Subgroups
CentraliserOfInvolution(G, g : parameters): GrpMat, GrpMatElt → GrpMatCentraliserOfInvolution(G, g, w : parameters): GrpMat, GrpMatElt, GrpSLPElt → GrpMat, []AreInvolutionsConjugate(G, x, wx, y, wy : parameters): GrpMat, GrpMatElt, GrpSLPElt, GrpMatElt, GrpSLPElt → BoolElt, GrpMatElt, GrpSLPEltNormalClosureMonteCarlo(G, H ): GrpMat, GrpMat → GrpMatNormalClosureMonteCarlo(G, H : parameters): GrpPerm, GrpPerm → GrpMatDerivedGroupMonteCarlo(G : parameters): GrpMat → GrpMatIsProbablyPerfect(G : parameters): Grp → BoolElt- Example: Testing a Subgroup for Perfection
- Aschbacher Reduction
- Constructive Recognition for Simple Groups
- Constructive Recognition for Classical Groups
ClassicalStandardGenerators(type, d, q): MonStgElt, RngIntElt, RngIntElt → []ClassicalConstructiveRecognition(G, type, d, q): GrpMat[FldFin], MonStgElt, RngIntElt, RngIntElt → BoolElt, Map, Map, Map, Map, SeqEnum, SeqEnumClassicalConstructiveRecognition(G): GrpMat[FldFin] → BoolElt, SeqEnum, SeqEnum- Example: Constructive Recognition of Sp(6, 125)
ClassicalChangeOfBasis(G): GrpMat[FldFin] → GrpMatElt[FldFin]ClassicalRewrite(G, gens, type, dim, q, g : parameters): Grp, SeqEnum, MonStgElt, RngIntElt, RngIntElt, GrpElt → BoolElt, GrpEltClassicalRewriteNatural(type, CB, g): MonStgElt, GrpMatElt, GrpMatElt → BoolElt, GrpEltClassicalRewriteNatural(G, type, CB, g): MonStgElt, GrpMatElt, GrpMatElt → BoolElt, GrpEltClassicalStandardPresentation(type, d, q : parameters): MonStgElt, RngIntElt, RngIntElt → SLPGroup, []- Example: Standard Generators for SL(6, 125)
- Constructive Recognition for Exceptional Groups
ExceptionalStandardGenerators(type, rank, q): MonStgElt, RngIntElt, RngIntElt → []ExceptionalConstructiveRecognition(G, type, rank, q): GrpMat[FldFin], MonStgElt, RngIntElt, RngIntElt → BoolElt, Map, Map, Map, Map, SeqEnum, SeqEnum- Example: Constructive Recognition of an Exceptional Group
ExceptionalRewrite(type, rank, q, X, Xm, g): MonStgElt, RngIntElt, RngIntElt, SeqEnum, SeqEnum, GrpElt → BoolElt, GrpEltExceptionalStandardPresentation(type, rank, q): MonStgElt, RngIntElt, RngIntElt → SLPGroup, []- Example: Standard Presentation of G2
- Composition Trees for Matrix Groups
- The Composition Tree Algorithm
- Constructing the Composition Tree
- Accessing the Composition Tree
CompositionTreeOrder(G): Grp → RngIntEltCompositionTreeNonAbelianFactors(G): Grp → RngIntEltDisplayCompTreeNodes(G : parameters): Grp- Example: Subgroup Order and Membership in J4
CompositionTreeNiceGroup(G): Grp → GrpMat[FldFin]CompositionTreeSLPGroup(G): Grp → GrpSLP, MapCompositionTreeNiceToUser(G): Grp → Map, []CompositionTreeOrder(G): Grp → RngIntEltCompositionTreeFactoredOrder(G): Grp → RngIntEltFactCompositionTreeElementToWord(G, g): Grp, GrpElt → BoolElt, GrpSLPEltCompositionTreeNonAbelianFactors(G): GrpMat[FldFin] → ListCompositionTreeCBM(G): GrpMat[FldFin] → GrpMatEltCompositionTreeReductionInfo(G, t): Grp, RngIntElt → MonStgElt, Grp, GrpCompositionTreeSeries(G): Grp → SeqEnum, List, List, List, BoolElt, []CompositionTreeFactorNumber(G, g): Grp, GrpElt → RngIntEltHasCompositionTree(G): Grp → BoolEltCleanCompositionTree(G): Grp- Example: Composition Tree of a Conformal Orthogonal Group
- Example: Composition Tree of a Maximal Subgroup
- The LMG functions
SetLMGSchreierBound(n): RngIntEltLMGInitialize(G : parameters): GrpMatLMGInitialise(G : parameters): GrpMatLMGOrder(G): GrpMat[FldFin] → RngIntEltLMGFactoredOrder(G): GrpMat[FldFin] → SeqEnumLMGIsIn(G, x): GrpMat, GrpMatElt → BoolElt, GrpSLPEltLMGIsSubgroup(G, H): GrpMat, GrpMat → BoolEltLMGEqual(G, H): GrpMat, GrpMat → BoolEltLMGIndex(G, H): GrpMat, GrpMat → RngIntEltLMGIsNormal(G, H): GrpMat, GrpMat → BoolEltLMGNormalClosure(G, H): GrpMat, GrpMat → GrpMatLMGDerivedGroup(G): GrpMat → GrpMatLMGCommutatorSubgroup(G, H): GrpMat, GrpMat → GrpMatLMGIsSoluble(G): GrpMat → BoolEltLMGIsSolvable(G): GrpMat → BoolEltLMGIsNilpotent(G): GrpMat → BoolEltLMGCompositionSeries(G): GrpMat[FldFin] → SeqEnumLMGCompositionFactors(G): GrpMat[FldFin] → SeqEnumLMGChiefSeries(G): GrpMat[FldFin] → SeqEnumLMGChiefFactors(G): GrpMat[FldFin] → SeqEnumLMGUnipotentRadical(G): GrpMat → GrpMat, GrpPC, MapLMGSolubleRadical(G): GrpMat → GrpMat, GrpPC, MapLMGSolvableRadical(G): GrpMat → GrpMat, GrpPC, MapLMGFittingSubgroup(G): GrpMat → GrpMat, GrpPC, MapLMGCentre(G): GrpMat → GrpMatLMGCenter(G): GrpMat → GrpMatLMGSylow(G,p): GrpMat, RngIntElt → GrpMatLMGSocleStar(G): GrpMat → GrpMatLMGSocleStarFactors(G): GrpMat → SeqEnum, SeqEnumLMGSocleStarAction(G): GrpMat → Map, GrpPerm, GrpMatLMGSocleStarActionKernel(G): GrpMat → GrpMat, GrpPC, MapLMGSocleStarQuotient(G): GrpMat → GrpPerm, Map, GrpMat- Example: LMG Functions on a Maximal Subgroup
LMGRadicalQuotient(G): GrpMat → GrpPerm, Map, GrpMatLMGCentraliser(G, g): GrpMat, GrpMatElt → GrpMatLMGCentralizer(G, g): GrpMat, GrpMatElt → GrpMatLMGIsConjugate(G, g, h): GrpMat, GrpMatElt, GrpMatElt → BoolElt, GrpMatEltLMGClasses(G): GrpMat → SeqEnumLMGConjugacyClasses(G): GrpMat → SeqEnumLMGNormaliser(G, H): GrpMat, GrpMat → GrpMatLMGNormalizer(G, H): GrpMat, GrpMat → GrpMatLMGIsConjugate(G, H, K): GrpMat, GrpMat, GrpMat → BoolElt, GrpMatEltLMGMeet(G, H, K): GrpMat, GrpMat, GrpMat → GrpMatLMGMaximalSubgroups(G): GrpMat → SeqEnumLMGNormalSubgroups(G): GrpMat → SeqEnumLMGLowIndexSubgroups(G,n): GrpMat, RngIntElt → SeqEnumLMGCosetAction(G,H : parameters): GrpMat, GrpMat → Map, GrpPerm, GrpMatLMGCosetImage(G,H): GrpMat, GrpMat → GrpPermLMGCosetActionInverseImage(G, f, i): GrpMat, Map, RngIntElt → GrpMatEltLMGRightTransversal(G,H : parameters): GrpMat, GrpMat → SeqEnumLMGIsPrimitive(G): GrpMat → BoolEltLMGCharacterTable(G : parameters): GrpMat → SeqEnum- Example: Classes and Characters of the Dempwolff Group
- Finding a Base
- Unipotent Matrix Groups
- Bibliography