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ALMOST SIMPLE GROUPS

 
Acknowledgements
 
Introduction
      Overview
 
Creating Finite Groups of Lie Type
      Generic Creation Function
      The Orders of the Chevalley Groups
      Classical Groups
            Linear Groups
            Unitary Groups
            Symplectic Groups
            Orthogonal and Spin Groups
      Exceptional Groups
            Suzuki Groups
            Small Ree Groups
            Large Ree Groups
 
Group Recognition
      Constructive Recognition of Alternating Groups
      Determining the Type of a Finite Group of Lie Type
      Classical Forms
      Recognizing Classical Groups in their Natural Representation
      Constructive Recognition of Linear Groups
      Constructive Recognition of Symplectic Groups
      Constructive Recognition of Unitary Groups
      Constructive Recognition of SL(d, q) in Low Degree
      Constructive Recognition of Suzuki Groups
            Introduction
            Recognition Functions
      Constructive Recognition of Small Ree Groups
            Introduction
            Recognition Functions
      Constructive Recognition of Large Ree Groups
            Introduction
            Recognition Functions
 
Properties of Finite Groups Of Lie Type
      Maximal Subgroups of the Classical Groups
      Maximal Subgroups of the Exceptional Groups
      Sylow Subgroups of the Classical Groups
      Sylow Subgroups of Exceptional Groups
      Conjugacy of Subgroups of the Classical Groups
      Conjugacy of Elements of the Exceptional Groups
      Irreducible Subgroups of the General Linear Group
 
Atlas Data for the Sporadic Groups
 
Bibliography







DETAILS

 
Introduction

      Overview

 
Creating Finite Groups of Lie Type

      Generic Creation Function
            ChevalleyGroup(X, n, K: parameters) : MonStgElt, RngIntElt, FldFin -> GrpMat

      The Orders of the Chevalley Groups
            ChevalleyOrderPolynomial(type, n: parameters) : MonStgElt, RngIntElt -> RngUPolElt
            FactoredChevalleyGroupOrder(type, n, F: parameters) : MonStgElt, RngIntElt, FldFin -> RngIntEltFact

      Classical Groups

            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

            Unitary Groups
                  ConformalUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  GeneralUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpecialUnitaryGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

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

            Orthogonal and Spin Groups
                  ConformalOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  GeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  ConformalOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  GeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  ConformalOrthogonalGroupMinus(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
                  Spin(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpinPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpinMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

      Exceptional Groups

            Suzuki Groups
                  SuzukiGroup(q) : RngIntElt -> GrpMat
                  Example GrpASim_Symplectic (H65E1)
                  Example GrpASim_Suzuki (H65E2)

            Small Ree Groups
                  ReeGroup(q) : RngIntElt -> GrpMat

            Large Ree Groups
                  LargeReeGroup(q) : RngIntElt -> GrpMat

 
Group Recognition

      Constructive Recognition of Alternating Groups
            RecogniseAlternatingOrSymmetric(G, n) : Grp, RngIntElt -> BoolElt, BoolElt, UserProgram, UserProgram
            Example GrpASim_RecogniseAltsym1 (H65E3)
            RecogniseSymmetric(G, n: parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
            SymmetricElementToWord (G, g) : Grp, GrpElt -> BoolElt, GrpSLPElt
            RecogniseAlternating(G, n: parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
            AlternatingElementToWord (G, g) : Grp, GrpElt -> BoolElt, GrpSLPElt
            GuessAltsymDegree(G: parameters) : Grp -> BoolElt, MonStgElt, RngIntElt
            Example GrpASim_RecogniseAltsym2 (H65E4)

      Determining the Type of a Finite Group of Lie Type
            LieCharacteristic(G : parameters) : Grp -> RngIntElt
            Example GrpASim_WriteOverSmallerField (H65E5)
            LieType(G, p : parameters) : GrpMat, RngIntElt -> BoolElt, Tup
            SimpleGroupName(G : parameters): GrpMat -> BoolElt, List
            Example GrpASim_IdentifySimple (H65E6)

      Classical Forms
            ClassicalForms(G: parameters): GrpMat -> Rec
            SymplecticForm(G: parameters) : GrpMat -> BoolElt, AlgMatElt [,SeqEnum]
            SymmetricBilinearForm(G: parameters) : GrpMat -> BoolElt, AlgMatElt, MonStgElt [,SeqEnum]
            QuadraticForm(G): GrpMat -> BoolElt, AlgMatElt, MonStgElt [,SeqEnum]
            UnitaryForm(G) : GrpMat -> BoolElt, AlgMatElt [,SeqEnum]
            FormType(G) : GrpMat -> MonStgElt
            Example GrpASim_ClassicalForms (H65E7)
            TransformForm(form, type) : AlgMatElt, MonStgElt -> GrpMatElt
            TransformForm(G) : GrpMat -> GrpMatElt
            SpinorNorm(g, form): GrpMatElt, AlgMatElt -> RngIntElt

      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 GrpASim_RecognizeClassical (H65E8)

      Constructive Recognition of Linear Groups
            RecognizeSL2(G) : GrpMat -> BoolElt, Map, Map, Map, Map
            SL2ElementToWord(G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
            SL2Characteristic(G : parameters) : GrpMat -> RngIntElt, RngIntElt
            Example GrpASim_RecognizeSL2-1 (H65E9)
            Example GrpASim_RecogniseSL2-2 (H65E10)
            RecogniseSL3(G) : GrpMat -> BoolElt, Map, Map, Map, Map
            SL3ElementToWord (G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
            Example GrpASim_RecogniseSL3 (H65E11)
            RecogniseSL(G, d, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map

      Constructive Recognition of Symplectic Groups
            RecogniseSpOdd(G, d, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map
            RecogniseSp4Even(G, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map

      Constructive Recognition of Unitary Groups
            RecogniseSU3(G, d, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map
            RecogniseSU4(G, d, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map

      Constructive Recognition of SL(d, q) in Low Degree
            RecogniseSymmetricSquare (G) : GrpMat -> BoolElt, GrpMat
            SymmetricSquarePreimage (G, g) : GrpMat, GrpMatElt -> GrpMatElt
            RecogniseAlternatingSquare (G) : GrpMat -> BoolElt, GrpMat
            AlternatingSquarePreimage (G, g) : GrpMat, GrpMatElt -> GrpMatElt
            RecogniseAdjoint (G) : GrpMat -> BoolElt, GrpMat
            AdjointPreimage (G, g) : GrpMat, GrpMatElt -> GrpMatElt
            RecogniseDelta (G) : GrpMat -> BoolElt, GrpMat
            DeltaPreimage (G, g) : GrpMat, GrpMatElt -> GrpMatElt
            Example GrpASim_RecogniseSymmetricSquare (H65E12)

      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
                  SzPresentation(q) : RngIntElt -> GrpFP, HomGrp
                  SatisfiesSzPresentation(G) : GrpMat -> BoolElt
                  SuzukiIrreducibleRepresentation(F, twists : parameters) : FldFin, SeqEnum[RngIntElt] -> GrpMat
                  Example GrpASim_ex-1 (H65E13)
                  Example GrpASim_ex-2 (H65E14)
                  Example GrpASim_ex-3 (H65E15)
                  Example GrpASim_ex-4 (H65E16)

      Constructive Recognition of Small Ree Groups

            Introduction

            Recognition Functions
                  RecogniseRee(G : parameters) : GrpMat -> BoolElt, Map, Map, Map, Map
                  ReeElementToWord(G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
                  IsReeGroup(G) : GrpMat -> BoolElt, RngIntElt
                  ReeIrreducibleRepresentation(F, twists : parameters) : FldFin, SeqEnum[RngIntElt] -> GrpMat
                  Example GrpASim_ex-1 (H65E17)

      Constructive Recognition of Large Ree Groups

            Introduction

            Recognition Functions
                  RecogniseLargeRee(G : parameters) : GrpMat -> BoolElt, Map, Map, Map, Map
                  LargeReeElementToWord(G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
                  IsLargeReeGroup(G) : GrpMat -> BoolElt, RngIntElt

 
Properties of Finite Groups Of Lie Type

      Maximal Subgroups of the Classical Groups
            ClassicalMaximals(type, d, q : parameters) : MonStgElt, RngIntElt, RngIntElt -> SeqEnum

      Maximal Subgroups of the Exceptional Groups
            SuzukiMaximalSubgroups(G) : GrpMat -> SeqEnum, SeqEnum
            SuzukiMaximalSubgroupsConjugacy(G, R, S) : GrpMat, GrpMat, GrpMat -> GrpMatElt, GrpSLPElt
            ReeMaximalSubgroups(G) : GrpMat -> SeqEnum, SeqEnum
            ReeMaximalSubgroupsConjugacy(G, R, S) : GrpMat, GrpMat, GrpMat -> GrpMatElt, GrpSLPElt

      Sylow Subgroups of the Classical Groups
            ClassicalSylow(G,p) : GrpMat, RngIntElt -> GrpMat
            ClassicalSylowConjugation(G,P,S) : GrpMat, GrpMat, GrpMat -> GrpMatElt
            ClassicalSylowNormaliser(G,P) : GrpMat, GrpMat -> GrpMatElt
            ClassicalSylowToPC(G,P) : GrpMat, GrpMat -> GrpPC, UserProgram, Map
            Example GrpASim_sylow_ex (H65E18)

      Sylow Subgroups of Exceptional Groups
            SuzukiSylow(G, p) : GrpMat, RngIntElt -> GrpMat, SeqEnum
            SuzukiSylowConjugacy(G, R, S, p) : GrpMat, GrpMat, GrpMat, RngIntElt -> GrpMatElt, GrpSLPElt
            Example GrpASim_sz-sylow (H65E19)
            ReeSylow(G, p) : GrpMat, RngIntElt -> GrpMat, SeqEnum
            ReeSylowConjugacy(G, R, S, p) : GrpMat, GrpMat, GrpMat, RngIntElt -> GrpMatElt, GrpSLPElt
            LargeReeSylow(G, p) : GrpMat, RngIntElt -> GrpMat, SeqEnum
            Example GrpASim_ree-sylow (H65E20)

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

      Conjugacy of Elements of the Exceptional Groups
            SzConjugacyClasses(G) : GrpMat -> SeqEnum
            SzClassRepresentative(G, g) : GrpMat, GrpMatElt -> GrpMatElt, GrpMatElt
            SzIsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt
            SzClassMap(G) : GrpMat -> Map
            ReeConjugacyClasses(G) : GrpMat -> SeqEnum

      Irreducible Subgroups of the General Linear Group
            IrreducibleSubgroups(n, q) : RngIntElt, RngIntElt -> SeqEnum
            IrreducibleSolubleSubgroups(n, q) : RngIntElt, RngIntElt -> SeqEnum
            Example GrpASim_WriteOverSmallerField (H65E21)

 
Atlas Data for the Sporadic Groups
      StandardGenerators(G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
      IsomorphismToStandardCopy(G, str : parameters) : Grp, MonStgElt -> BoolElt, Map
      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
      Example GrpASim_SporadicJ1 (H65E22)

 
Bibliography

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