[Next][Prev] [Right] [Left] [Up] [Index] [Root]

MATRIX GROUPS OVER FINITE FIELDS

 
Acknowledgements
 
Introduction
 
Finding Elements with Prescribed Properties
 
Monte Carlo Algorithms for Subgroups
 
Aschbacher Reduction
      Introduction
      Primitivity
      Semilinearity
      Tensor Products
      Tensor-induced Groups
      Normalisers of Extraspecial r-groups and Symplectic 2-groups
      Writing Representations over Subfields
      Decompositions with Respect to a Normal Subgroup
            Accessing the Decomposition Information
 
Constructive Recognition for Simple Groups
 
Composition Trees for Matrix Groups
 
The LMG functions
 
Unipotent Matrix Groups
 
Bibliography







DETAILS

 
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 (H60E1)

 
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 (H60E2)

      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 (H60E3)

      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 (H60E4)

      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 (H60E5)

      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 (H60E6)

      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 (H60E7)
            WriteOverSmallerField(G, F) : GrpMat, FldFin -> GrpMat, Map
            Example GrpMatFF_WriteOverSmallerField (H60E8)

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

            Accessing the Decomposition Information
                  Example GrpMatFF_Decompose (H60E9)

 
Constructive Recognition for Simple Groups
      ClassicalStandardGenerators(type, d, q) : MonStgElt, RngIntElt, RngIntElt -> []
      ClassicalConstructiveRecognition(G : parameters) : GrpMat[FldFin] -> BoolElt, [], [], GrpMatElt
      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 (H60E10)

 
Composition Trees for Matrix Groups
      CompositionTree(G : parameters) : GrpMat[FldFin] -> []
      CompositionTreeFastVerification(G) : Grp -> BoolElt
      CompositionTreeVerify(G) : Grp -> BoolElt, []
      CompositionTreeNiceGroup(G) : Grp -> GrpMat[FldFin]
      CompositionTreeSLPGroup(G) : Grp -> GrpSLP, Map
      DisplayCompTreeNodes(G : parameters) : Grp ->
      CompositionTreeNiceToUser(G) : Grp -> Map, []
      CompositionTreeOrder(G) : Grp -> RngIntElt
      CompositionTreeElementToWord(G, g) : Grp, GrpElt -> BoolElt, GrpSLPElt
      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 (H60E11)
      Example GrpMatFF_CompTree2 (H60E12)

 
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
      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 (H60E13)
      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
      LMGMaximalSubgroups(G) : GrpMat -> SeqEnum

 
Unipotent Matrix Groups
      UnipotentMatrixGroup(G) : GrpMat -> GrpMatUnip
      WordMap(G) : GrpMatUnip -> Map
      Example GrpMatFF_UnipPCWordMap (H60E14)
      PCPresentation(G) : GrpMatUnip -> GrpPC, Map, Map
      Order(G) : GrpMatUnip -> RngIntElt
      g in G : GrpMatElt, GrpMatUnip -> BoolElt
      Example GrpMatFF_UnipPCPres (H60E15)

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Wed Apr 24 15:09:57 EST 2013