Introduction

 
Finding Elements with Prescribed Properties
      RandomElementOfOrder(G, n : parameters) : GrpMat, RngIntElt-> BoolElt, GrpMatElt, GrpSLPElt, BoolElt
      RandomElementOfNormalClosure(G, N): Grp -> GrpElt
      InvolutionClassicalGroupEven(G : parameters) : GrpMat[FldFin] ->GrpMatElt[FldFin], GrpSLPElt, RngIntElt

 
Monte Carlo Algorithms for Subgroups
      CentraliserOfInvolution(G, g : parameters) : GrpMat,GrpMatElt -> GrpMat
      CentraliserOfInvolution(G, g, w : parameters) : GrpMat,GrpMatElt, GrpSLPElt -> GrpMat, []
      AreInvolutionsConjugate(G, x, wx, y, wy : parameters) : GrpMat,GrpMatElt, GrpSLPElt, GrpMatElt, GrpSLPElt -> BoolElt, GrpMatElt, GrpSLPElt
      NormalClosureMonteCarlo(G, H ) : GrpMat, GrpMat -> GrpMat
      DerivedGroupMonteCarlo(G : parameters) : GrpMat -> GrpMat
      IsProbablyPerfect(G : parameters): Grp -> BoolElt
      Example GrpMatFF_IsProbablyPerfect (H67E1)

 
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 (H67E2)

      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 (H67E3)

      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 (H67E4)

      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 (H67E5)

      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 (H67E6)

      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 (H67E7)
            WriteOverSmallerField(G, F) : GrpMat, FldFin -> GrpMat, Map
            Example GrpMatFF_WriteOverSmallerField (H67E8)

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

            Accessing the Decomposition Information
                  Example GrpMatFF_Decompose (H67E9)

 
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, SeqEnum
            Example GrpMatFF_ClassicalConstructiveRecognition (H67E10)
            ClassicalChangeOfBasis(G): GrpMat[FldFin] -> GrpMatElt[FldFin]
            ClassicalRewrite(G, gens, type, dim, q, g : parameters): Grp, SeqEnum, MonStgElt, RngIntElt, RngIntElt, GrpElt -> BoolElt, GrpElt
            ClassicalRewriteNatural(type, CB, g): MonStgElt, GrpMatElt, GrpMatElt-> BoolElt, GrpElt
            ClassicalStandardPresentation (type, d, q : parameters) : MonStgElt, RngIntElt, RngIntElt -> SLPGroup, []
            Example GrpMatFF_ClassicalConstructiveRecognition (H67E11)

      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 GrpMatFF_ExceptionalConstructiveRecognition (H67E12)
            ExceptionalRewrite(type, rank, q, X, Xm, g): MonStgElt, RngIntElt, RngIntElt,SeqEnum, SeqEnum, GrpElt -> BoolElt, GrpElt
            ExceptionalStandardPresentation (type, rank, q) : MonStgElt, RngIntElt, RngIntElt -> SLPGroup, []
            Example GrpMatFF_exp_standard (H67E13)

 
Composition Trees for Matrix Groups

      The Composition Tree Algorithm

      Constructing the Composition Tree
            CompositionTree(G) : GrpMat[FldFin] -> []
            Example GrpMatFF_CompositionTree (H67E14)
            CompositionTree(G : parameters) : GrpMat[FldFin] -> []
            CompositionTreeFastVerification(G) : Grp -> BoolElt
            CompositionTreeVerify(G) : Grp -> BoolElt, []
            Example GrpMatFF_CompositionTreeVerify (H67E15)

      Accessing the Composition Tree
            CompositionTreeOrder(G) : Grp -> RngIntElt
            CompositionTreeNonAbelianFactors(G) : Grp -> RngIntElt
            DisplayCompTreeNodes(G : parameters) : Grp ->
            Example GrpMatFF_CompTreeJ4 (H67E16)
            CompositionTreeNiceGroup(G) : Grp -> GrpMat[FldFin]
            CompositionTreeSLPGroup(G) : Grp -> GrpSLP, Map
            CompositionTreeNiceToUser(G) : Grp -> Map, []
            CompositionTreeOrder(G) : Grp -> RngIntElt
            CompositionTreeElementToWord(G, g) : Grp, GrpElt -> BoolElt, GrpSLPElt
            CompositionTreeNonAbelianFactors(G) : GrpMat[FldFin] -> List
            CompositionTreeCBM(G) : GrpMat[FldFin -> GrpMatElt
            CompositionTreeReductionInfo(G, t) : Grp, RngIntElt -> MonStgElt,Grp, Grp
            CompositionTreeSeries(G) : Grp -> SeqEnum, List, List, List, BoolElt, []
            CompositionTreeFactorNumber(G, g) : Grp, GrpElt -> RngIntElt
            HasCompositionTree(G) : Grp -> BoolElt
            CleanCompositionTree(G) : Grp ->
            Example GrpMatFF_CompTree1 (H67E17)
            Example GrpMatFF_CompTree2 (H67E18)

 
The LMG functions
      SetLMGSchreierBound(n) : RngIntElt ->
      LMGInitialize(G : parameters) : GrpMat ->
      LMGOrder(G) : GrpMat[FldFin] -> RngIntElt
      LMGFactoredOrder(G) : GrpMat[FldFin] -> SeqEnum
      LMGIsIn(G, x) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
      LMGIsSubgroup(G, H) : GrpMat, GrpMat -> BoolElt
      LMGEqual(G, H) : GrpMat, GrpMat -> BoolElt
      LMGIndex(G, H) : GrpMat, GrpMat -> RngIntElt
      LMGIsNormal(G, H) : GrpMat, GrpMat -> BoolElt
      LMGNormalClosure(G, H) : GrpMat, GrpMat -> GrpMat
      LMGDerivedGroup(G) : GrpMat -> GrpMat
      LMGCommutatorSubgroup(G, H) : GrpMat, GrpMat -> GrpMat
      LMGIsSoluble(G) : GrpMat -> BoolElt
      LMGIsNilpotent(G) : GrpMat -> BoolElt
      LMGCompositionSeries(G) : GrpMat[FldFin] -> SeqEnum
      LMGCompositionFactors(G) : GrpMat[FldFin] -> SeqEnum
      LMGChiefSeries(G) : GrpMat[FldFin] -> SeqEnum
      LMGChiefFactors(G) : GrpMat[FldFin] -> SeqEnum
      LMGUnipotentRadical(G) : GrpMat -> GrpMat, GrpPC, Map
      LMGSolubleRadical(G) : GrpMat -> GrpMat, GrpPC, Map
      LMGFittingSubgroup(G) : GrpMat -> GrpMat, GrpPC, Map
      LMGCentre(G) : GrpMat -> GrpMat
      LMGSylow(G,p) : GrpMat, RngIntElt -> GrpMat
      LMGSocleStar(G) : GrpMat -> GrpMat
      LMGSocleStarFactors(G) : GrpMat -> SeqEnum, SeqEnum
      LMGSocleStarAction(G) : GrpMat -> Map, GrpPerm, GrpMat
      LMGSocleStarActionKernel(G) : GrpMat -> GrpMat, GrpPC, Map
      LMGSocleStarQuotient(G) : GrpMat -> GrpPerm, Map, GrpMat
      Example GrpMatFF_LMGex (H67E19)
      LMGRadicalQuotient(G) : GrpMat -> GrpPerm, Map, GrpMat
      LMGCentraliser(G, g) : GrpMat, GrpMatElt -> GrpMat
      LMGIsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt
      LMGClasses(G) : GrpMat -> SeqEnum
      LMGNormaliser(G, H) : GrpMat, GrpMat -> GrpMat
      LMGIsConjugate(G, H, K) : GrpMat, GrpMat, GrpMat -> BoolElt, GrpMatElt
      LMGMeet(G, H, K) : GrpMat, GrpMat, GrpMat -> GrpMat
      LMGMaximalSubgroups(G) : GrpMat -> SeqEnum
      LMGNormalSubgroups(G) : GrpMat -> SeqEnum
      LMGLowIndexSubgroups(G,n) : GrpMat, RngIntElt -> SeqEnum
      LMGCosetAction(G,H : parameters) : GrpMat, GrpMat -> Map, GrpPerm, GrpMat
      LMGCosetImage(G,H) : GrpMat, GrpMat -> GrpPerm
      LMGCosetActionInverseImage(G, f, i) : GrpMat, Map, RngIntElt -> GrpMatElt
      LMGRightTransversal(G,H : parameters) : GrpMat, GrpMat -> SeqEnum
      LMGIsPrimitive(G) : GrpMat -> BoolElt
      LMGCharacterTable(G : parameters) : GrpMat -> SeqEnum
      Example GrpMatFF_LMGex2 (H67E20)

 
Finding a Base
      LMGBase(G : parameters) : GrpMat -> Tup, SeqEnum, SeqEnum
      Example GrpMatFF_base (H67E21)

 
Unipotent Matrix Groups
      UnipotentMatrixGroup(G) : GrpMat -> GrpMatUnip
      WordMap(G) : GrpMatUnip -> Map
      Example GrpMatFF_UnipPCWordMap (H67E22)
      PCPresentation(G) : GrpMatUnip -> GrpPC, Map, Map
      Order(G) : GrpMatUnip -> RngIntElt
      g in G : GrpMatElt, GrpMatUnip -> BoolElt
      Example GrpMatFF_UnipPCPres (H67E23)

 
Bibliography

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