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)
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
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013