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
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
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
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
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
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
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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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012