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Acknowledgements
Introduction
Overview
Creating Finite Groups of Lie Type
Generic Creation Function
Classical Groups
Linear Groups
Unitary Groups
Symplectic Groups
Orthogonal Groups
Exceptional Groups
Suzuki Groups
Small Ree Groups
Large Ree Groups
Group Recognition
Non-Constructive Recognition of Alternating Groups
Determining the Type of a Finite Group of Lie Type
Classical forms
Recognizing Classical Groups in their Natural Representation
Constructive Recognition of Linear Groups
Constructive Recognition of Suzuki Groups
Introduction
Recognition Functions
Constructive Recognition of Small Ree Groups
Introduction
Recognition Functions
Constructive Recognition of Large Ree Groups
Introduction
Recognition Functions
Properties of Finite Groups Of Lie Type
Maximal Subgroups of the Classical Groups
Maximal Subgroups of the Exceptional Groups
Sylow Subgroups of the Classical Groups
Sylow Subgroups of Exceptional Groups
Conjugacy of Subgroups of the Classical Groups
Conjugacy of elements of the Exceptional Groups
Irreducible Subgroups of the General Linear Group
Atlas Data for the Sporadic Groups
Bibliography
DETAILS
Introduction
Overview
Creating Finite Groups of Lie Type
Generic Creation Function
ChevalleyGroup(s, n, K: parameters) : MonStgElt, RngIntElt, FldFin -> GrpMat
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
Symplectic Groups
ConformalSymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
SymplecticGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
Orthogonal Groups
GeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
SpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
GeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
SpecialOrthogonalGroupPlus(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
Exceptional Groups
Suzuki Groups
SuzukiGroup(q) : RngIntElt -> GrpMat
Example GrpASim_Symplectic (H56E1)
Example GrpASim_Suzuki (H56E2)
Small Ree Groups
ReeGroup(q) : RngIntElt -> GrpMat
Large Ree Groups
LargeReeGroup(q) : RngIntElt -> GrpMat
Group Recognition
Non-Constructive Recognition of Alternating Groups
RecogniseAlternatingOrSymmetric(G, n) : Grp, RngIntElt -> BoolElt, BoolElt, UserProgram, UserProgram
Example GrpASim_RecogniseAltsym1 (H56E3)
RecogniseSymmetric(G, n: parameters) : Grp, RngIntElt -> BoolElt, BoolElt, UserProgram, UserProgram
RecogniseAlternating(G, n: parameters) : Grp, RngIntElt -> BoolElt, BoolElt, UserProgram, UserProgram
GuessAltsymDegree(G: parameters) : Grp -> BoolElt, MonStgElt, RngIntElt
Example GrpASim_RecogniseAltsym2 (H56E4)
Determining the Type of a Finite Group of Lie Type
LieCharacteristic(G : parameters) : Grp -> RngIntElt
Example GrpASim_WriteOverSmallerField (H56E5)
LieType(G, p : parameters) : GrpMat, RngIntElt -> BoolElt, Tup
SimpleGroupName(G : parameters): GrpMat -> BoolElt, List
Example GrpASim_IdentifySimple (H56E6)
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 (H56E7)
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 (H56E8)
Constructive Recognition of Linear Groups
RecognizeSL2(G, q : parameters) : GrpMat, RngIntElt -> BoolElt, Map, Map, Map, Map
SL2ElementToWord(G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
SL2Presentation(q : parameters) : RngIntElt -> GrpFP
SatisfiesSL2Presentation(G, q : parameters) : GrpMat, RngIntElt -> BoolElt
SL2Characteristic(G : parameters) : GrpMat -> RngIntElt, RngIntElt
Example GrpASim_RecogniseSL2-1 (H56E9)
Example GrpASim_RecogniseSL2-2 (H56E10)
RecognizeSL3(G: parameters) : GrpMat, RngIntElt -> BoolElt, Map, Map, Map, Map
SL3ElementToWord (G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
Example GrpASim_RecogniseSL3 (H56E11)
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 (H56E12)
Example GrpASim_ex-2 (H56E13)
Example GrpASim_ex-3 (H56E14)
Example GrpASim_ex-4 (H56E15)
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 (H56E16)
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
Sylow Subgroups of the Classical Groups
ClassicalSylow(G,type,p) : GrpMat, MonStgElt, RngIntElt -> GrpMat
SylowConjClassical(G,P,S,type,p) : GrpMat, GrpMat, GrpMat, MonStgElt, RngIntElt -> GrpMatElt
ClassicalSylowNormaliser(G,P,type,p) : GrpMat, GrpMat, MonStgElt, RngIntElt -> GrpMatElt
ClassicalSylowToPC(P,type,p) : GrpMat, MonStgElt, RngIntElt -> GrpPC, UserProgram, Map
Example GrpASim_sylow_ex (H56E17)
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 (H56E18)
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 (H56E19)
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 (H56E20)
Atlas Data for the Sporadic Groups
StandardGenerators (G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
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
Bibliography
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