Introduction

      Overview

 
Double Covers of the Alternating and Symmetric Groups
      DoubleCoverSymmetricGroup(n: parameters) : RngIntElt -> GrpMat
      DoubleCoverAlternatingGroup(n: parameters) : RngIntElt -> GrpMat

 
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
                  ConformalSpecialUnitaryGroup(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
                  ConformalSpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
                  ConformalOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  GeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  ConformalSpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  ConformalOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  GeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  SpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
                  ConformalSpecialOrthogonalGroupMinus(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 (H72E1)
                  Example GrpASim_Suzuki (H72E2)

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

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

 
Group Recognition

      Constructive Recognition of Alternating Groups
            RecogniseAlternatingOrSymmetric(G : parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
            AlternatingOrSymmetricElementToWord(G, g): Grp, GrpElt -> BoolElt, GrpSLPElt
            Example GrpASim_RecogniseAltsym2 (H72E3)
            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 (H72E4)

      Determining the Type of a Finite Group of Lie Type
            LieCharacteristic(G : parameters) : Grp -> RngIntElt
            Example GrpASim_WriteOverSmallerField (H72E5)
            LieType(G, p : parameters) : GrpMat, RngIntElt -> BoolElt, Tup
            SimpleGroupName(G : parameters): GrpMat -> BoolElt, List
            Example GrpASim_IdentifySimple (H72E6)

      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 (H72E7)
            TransformForm(form, type) : AlgMatElt, MonStgElt -> GrpMatElt
            TransformForm(G) : GrpMat -> GrpMatElt
            SpinorNorm(g, form): GrpMatElt, AlgMatElt -> RngIntElt
            Example GrpASim_Spinor (H72E8)

      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
            ClassicalGroupType(G) : GrpMat -> BoolElt, MonStgElt
            Example GrpASim_RecognizeClassical (H72E9)

      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 (H72E10)
            Example GrpASim_RecogniseSL2-2 (H72E11)
            RecogniseSL3(G) : GrpMat -> BoolElt, Map, Map, Map, Map
            SL3ElementToWord (G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
            Example GrpASim_RecogniseSL3 (H72E12)
            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
            RecogniseSp4(G, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map, Map, Map, SeqEnum, SeqEnum

      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

      Recognition Of Classical Groups in Low Degree
            RecogniseSmallDegree(G) : GrpMat -> BoolElt, GrpMat
            SmallDegreePreimage(G, g) : GrpMat, GrpMatElt -> GrpMatElt
            SmallDegreeImage(G, h) : GrpMat, GrpMatElt -> GrpMatElt
            Example GrpASim_RecogniseSmallDegree (H72E13)

      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 (H72E14)
                  Example GrpASim_ex-2 (H72E15)
                  Example GrpASim_ex-3 (H72E16)
                  Example GrpASim_ex-4 (H72E17)

      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 (H72E18)

      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
            SzMaximals(q) : RngIntElt -> SeqEnum
            ReeMaximals(q) : RngIntElt -> SeqEnum
            G2Maximals(q) : RngIntElt -> SeqEnum

      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 (H72E19)

      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 (H72E20)
            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 (H72E21)

      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
            Example GrpASim_WriteOverSmallerField (H72E22)

 
Atlas Data for the Sporadic Groups
      StandardGenerators(G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
      StandardGeneratorsGroupNames() : -> SetIndx
      StandardCopy(str) : MonStgElt -> Grp, BoolElt
      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 (H72E23)

 
Automorphism Groups of Finite Simple Groups
      AutomorphismGroupSimpleGroup(type, d, q) : MonStgElt, RngIntElt, RngIntElt -> GrpPerm

 
Bibliography

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