- Introduction
- Creating Finite Groups of Lie Type
- Generic Creation Function
- The Orders of the Chevalley Groups
- 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
- Symplectic Groups
- Orthogonal and Spin Groups
- ConformalOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
- GeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
- SpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
- ConformalSpecialOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
- ConformalOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
- GeneralOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
- SpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
- ConformalSpecialOrthogonalGroupPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
- ConformalOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
- GeneralOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
- SpecialOrthogonalGroupMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
- ConformalSpecialOrthogonalGroupMinus(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
- Exceptional Groups
- Group Recognition
- Constructive Recognition of Alternating Groups
- RecogniseAlternatingOrSymmetric(G : parameters) : Grp, RngIntElt -> BoolElt, Map, Map, Map, Map, BoolElt
- AlternatingOrSymmetricElementToWord(G, g): Grp, GrpElt -> BoolElt, GrpSLPElt
- Example GrpASim_RecogniseAltsym2 (H71E3)
- 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 (H71E4)
- Determining the Type of a Finite Group of Lie Type
- 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 (H71E7)
- TransformForm(form, type) : AlgMatElt, MonStgElt -> GrpMatElt
- TransformForm(G) : GrpMat -> GrpMatElt
- SpinorNorm(g, form): GrpMatElt, AlgMatElt -> RngIntElt
- Example GrpASim_Spinor (H71E8)
- Recognizing Classical Groups in their Natural Representation
- 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 (H71E10)
- Example GrpASim_RecogniseSL2-2 (H71E11)
- RecogniseSL3(G) : GrpMat -> BoolElt, Map, Map, Map, Map
- SL3ElementToWord (G, g) : GrpMat, GrpMatElt -> BoolElt, GrpSLPElt
- Example GrpASim_RecogniseSL3 (H71E12)
- 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
- RecogniseSp4(G, q) : Grp, RngIntElt, RngIntElt -> BoolElt, Map, Map, Map, Map, SeqEnum, SeqEnum
- 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
- Recognition Of Classical Groups in Low Degree
- 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 (H71E14)
- Example GrpASim_ex-2 (H71E15)
- Example GrpASim_ex-3 (H71E16)
- Example GrpASim_ex-4 (H71E17)
- 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 (H71E18)
- Constructive Recognition of Large Ree Groups
- Properties of Finite Groups Of Lie Type
- Maximal Subgroups of the Classical Groups
- 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
- SzMaximals(q) : RngIntElt -> SeqEnum
- ReeMaximals(q) : RngIntElt -> SeqEnum
- G2Maximals(q) : RngIntElt -> SeqEnum
- 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 (H71E19)
- 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 (H71E20)
- 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 (H71E21)
- 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
- StandardGenerators(G, str : parameters) : Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum
- StandardGeneratorsGroupNames() : -> SetIndx
- StandardCopy(str) : MonStgElt -> Grp, BoolElt
- 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 (H71E23)
- Automorphism Groups of Finite Simple Groups
- Bibliography
V2.28, 13 July 2023