MATRIX GROUPS OVER FINITE FIELDS

DETAILS

 
Introduction

      Overview

 
Monte-Carlo Functions
      NormalSubgroupRandomElement(G, N): Grp -> GrpElt
      IsProbablyPerfect(G : parameters): Grp -> BoolElt
      Example GrpMatFF_IsProbablyPerfect (H21E1)
      WriteOverSmallerField(G, F) : GrpMat, FldFin -> GrpMat, Map
      Example GrpMatFF_WriteOverSmallerField (H21E2)

 
Aschbacher Reduction

      Introduction

      Primitivity
            IsPrimitive(G: parameters) : GrpMat -> BoolElt
            ImprimitiveBasis (G) : GrpMat -> SeqEnum
            Blocks(G) : GrpMat -> SeqEnum
            BlocksImage(G) : GrpMat -> GrpPerm
            ImprimitiveAction(G, g) : GrpMat, GrpMatElt -> GrpPermElt
            Example GrpMatFF_IsPrimitive (H21E3)

      Semilinearity
            IsSemiLinear(G) : GrpMat -> BoolElt
            DegreeOfFieldExtension(G) : GrpMat -> RngIntElt
            CentralisingMatrix(G) : GrpMat -> AlgMatElt
            FrobeniusAutomorphisms(G) : GrpMat -> SeqEnum
            WriteOverLargerField(G) : GrpMat -> GrpMat, GrpAb, SeqEnum
            Example GrpMatFF_Semilinearity (H21E4)

      Tensor Products
            IsTensor(G: parameters) : GrpMat -> BoolElt
            TensorBasis(G) : GrpMat -> GrpMatElt
            TensorFactors(G) : GrpMat -> GrpMat, GrpMat
            IsProportional(X, k) : Mtrx, RngIntElt -> BoolElt, Tup
            Example GrpMatFF_Tensor (H21E5)

      Tensor-induced Groups
            IsTensorInduced(G : parameters) : GrpMat -> BoolElt
            TensorInducedBasis(G) : GrpMat -> GrpMatElt
            TensorInducedPermutations(G) : GrpMat -> SeqEnum
            TensorInducedAction(G, g) : GrpMat, GrpMatElt -> GrpPermElt
            Example GrpMatFF_TensorInduced (H21E6)

      Normalisers of Extraspecial r-groups and Symplectic 2-groups
            IsExtraSpecialNormaliser(G) : GrpMat -> BoolElt
            ExtraSpecialParameters(G) : GrpMat -> [RngIntElt, RngIntElt]
            ExtraSpecialGroup(G) : GrpMat -> GrpMat
            ExtraSpecialNormaliser(G) : GrpMat -> SeqEnum
            ExtraSpecialAction(G, g) : GrpMat, GrpMatElt -> GrpMatElt
            ExtraSpecialBasis(G) : GrpMat -> GrpMatElt
            Example GrpMatFF_ExtraSpecialNormaliser (H21E7)

      Writing Representations over Subfields
            IsOverSmallerField (G: parameters) : GrpMat -> BoolElt, GrpMat
            IsOverSmallerField (G, k: parameters) : GrpMat -> BoolElt, GrpMat
            SmallerField(G) : GrpMat -> FLdFin
            SmallerFieldBasis (G) : GrpMat -> GrpMatElt
            SmallerFieldImage (G, g) : GrpMat, GrpMatElt -> GrpMatElt
            Example GrpMatFF_IsOverSmallerField (H21E8)

      Decompositions with Respect to a Normal Subgroup
            SearchForDecomposition(G, S) : GrpMat, [GrpMatElt] -> BoolElt

            Accessing the Decomposition Information
                  Example GrpMatFF_Decompose (H21E9)

 
Creating Finite Groups of Lie Type
      ChevalleyGroup(s, n, K: parameters) : MonStgElt, RngIntElt, FldFin -> GrpMat

      Classical Groups

            General and Special Linear Groups
                  GeneralLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpecialLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  AffineGeneralLinearGroup(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat
                  AffineSpecialLinearGroup(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat

      General and Special Unitary Groups
            GeneralUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
            SpecialUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

      Symplectic Groups
            SymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

      Orthogonal Groups
            GeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
            SpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
            GeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
            SpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
            GeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
            SpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
            Omega(n, q) : RngIntElt, RngIntElt -> GrpMat
            OmegaPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
            OmegaMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

      Exceptional Groups

            Suzuki Groups
                  SuzukiGroup(q) : RngIntElt -> GrpMat
                  Example GrpMatFF_Symplectic (H21E10)
                  Example GrpMatFF_Suzuki (H21E11)

            Ree Groups
                  ReeGroup(q) : RngIntElt -> GrpMat

 
Group Recognition

      Determining the Type of a Finite Group of Lie Type
            LieCharacteristic(G : parameters) : Grp -> RngIntElt
            Example GrpMatFF_WriteOverSmallerField (H21E12)
            LieType(G, p : parameters) : GrpMat, RngIntElt -> BoolElt, Tup
            SimpleGroupName(G : parameters): GrpMat -> BoolElt, List
            Example GrpMatFF_IdentifySimple (H21E13)

      Classical forms
            ClassicalForms(G: parameters): GrpMat -> Rec
            SymplecticForm(G: parameters) : GrpMat -> AlgMatElt
            SymmetricBilinearForm(G: parameters) : GrpMat -> BoolElt, AlgMatElt, RngIntElt, SeqEnum
            QuadraticForm(G): GrpMat -> BoolElt, AlgMatElt, MonStgElt [,SeqEnum]
            UnitaryForm(G) : GrpMat -> BoolElt, AlgMatElt [,SeqEnum]
            FormType(G) : GrpMat -> MonStgElt
            Example GrpMatFF_ClassicalForms (H21E14)
            TransformForm(form, d, q, type) : AlgMatElt, RngIntElt, RngIntElt, MonStgElt -> GrpMatElt
            TransformForm(G) : GrpMat -> GrpMatElt

      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 GrpMatFF_RecognizeClassical (H21E15)

      Constructive Recognition of Linear Groups
            RecognizeSL2(G, q : parameters) : GrpMat, RngIntElt -> BoolElt, Map, Map, Map, Map
            SL2ElementToWord(G, g) : GrpMat, GrpMatElt ->
            SL2Presentation(q : parameters) : RngIntElt -> GrpFP
            SatisfiesSL2Presentation(G, q : parameters) : GrpMat, RngIntElt -> BoolElt
            SL2Characteristic(G) : GrpMat -> RngIntElt, RngIntElt
            Example GrpMatFF_RecogniseSL2-1 (H21E16)
            Example GrpMatFF_RecogniseSL2-2 (H21E17)
            RecognizeSL3(G: parameters) : GrpMat, RngIntElt -> BoolElt, Map, Map, Map, Map
            SL3ElementToWord (G, g) : GrpMat, GrpMatElt -> GrpSLPElt
            Example GrpMatFF_RecogniseSL3 (H21E18)

      Constructive Recognition of Suzuki Groups

            Introduction

            Recognition Functions
                  IsSuzukiGroup(G) : GrpMat -> BoolElt, RngIntElt
                  RecogniseSz(G : parameters) : GrpMat -> BoolElt, Map, Map, Map, Map
                  SzElementToWord(G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
                  PresentationOfSimpleGroup("Sz", q) : RngIntElt -> GrpFP, HomGrp
                  SatisfiesSzPresentation(G) : GrpMat -> BoolElt
                  SuzukiIrreducibleRepresentation(F, twists : parameters) : FldFin, SeqEnum[RngIntElt] -> GrpMat
                  Example GrpMatFF_ex-1 (H21E19)
                  Example GrpMatFF_ex-2 (H21E20)
                  Example GrpMatFF_ex-3 (H21E21)
                  Example GrpMatFF_ex-4 (H21E22)

      Constructive Recognition of Ree Groups

            Introduction

            Recognition Functions
                  RecogniseRee(G : parameters) : GrpMat -> BoolElt, Map, Map, Map, Map
                  ReeElementToWord(G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
                  ReeRecognition(G) : GrpMat -> BoolElt, RngIntElt
                  ReeIrreducibleRepresentation(F, twists : parameters) : FldFin, SeqEnum[RngIntElt] -> GrpMat
                  Example GrpMatFF_ex-1 (H21E23)

 
Properties of Finite Groups Of Lie Type

      Sylow Subgroups of the Classical Groups
            ClassicalSylow(G,type,p) : GrpMat, MonStgElt, RngIntElt -> GrpMat
            SylowConjClassical(G,P,S,type,p) : GrpMat, GrpMat, GrpMat, MonStgElt, RngIntElt -> GrpMatElt
            ClassicalSylowNormaliser(G,P,type,p) : GrpMat, GrpMat, MonStgElt, RngIntElt -> GrpMatElt
            ClassicalSylowToPC(P,type,p) : GrpMat, MonStgElt, RngIntElt -> GrpPC, UserProgram, Map
            Example GrpMatFF_sylow_ex (H21E24)

      Sylow Subgroups of Exceptional Groups
            SuzukiSylow(G, p) : GrpMat, RngIntElt -> GrpMat, SeqEnum
            SuzukiSylowConjugacy(G, R, S, p) : GrpMat, GrpMat, GrpMat, RngIntElt -> GrpMatElt, GrpSLPElt
            Example GrpMatFF_sz-sylow (H21E25)
            ReeSylow(G, p) : GrpMat, RngIntElt -> GrpMat, SeqEnum
            ReeSylowConjugacy(G, R, S, p) : GrpMat, GrpMat, GrpMat, RngIntElt -> GrpMatElt, GrpSLPElt
            Example GrpMatFF_ree-sylow (H21E26)

      Conjugacy of Subgroups of the Classical Groups
            IsGLConjugate(H, K) : GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass

      Irreducible Subgroups of the General Linear Group
            IrreducibleSubgroups(n, q) : RngIntElt, RngIntElt -> SeqEnum
            IrreducibleSolubleSubgroups(n, q) : RngIntElt, RngIntElt -> SeqEnum
            Example GrpMatFF_WriteOverSmallerField (H21E27)

 
Atlas Data for the Sporadic Groups
      StandardGenerators (G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
      StandardPresentation(G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
      MaximalSubgroups(G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
      Subgroups(G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum
      GoodBasePoints (G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum
      SubgroupsData (str) : MonStgElt -> SeqEnum
      MaximalSubgroupsData (str : parameters) : MonStgElt -> SeqEnum

 
Bibliography