Almost Simple Groups
- Introduction
- Double Covers of the Alternating and Symmetric Groups
- Creating Finite Groups of Lie Type
- Generic Creation Function
- The Orders of the Chevalley Groups
ChevalleyOrderPolynomial(type, n: parameters): MonStgElt, RngIntElt → RngUPolEltFactoredChevalleyGroupOrder(type, n, F: parameters): MonStgElt, RngIntElt, FldFin → RngIntEltFactFactoredChevalleyGroupOrder(type, n, q: parameters): MonStgElt, RngIntElt, RngIntElt → RngIntEltFactChevalleyGroupOrder(type, n, F: parameters): MonStgElt, RngIntElt, FldFin → RngIntEltFactChevalleyGroupOrder(type, n, q: parameters): MonStgElt, RngIntElt, RngIntElt → RngIntEltFact
- Classical Groups
- Linear Groups
GeneralLinearGroup(n, q): RngIntElt, RngIntElt → GrpMatGeneralLinearGroup(n, K): RngIntElt, FldFin → GrpMatGeneralLinearGroup(V): ModTupRng → GrpMatGL(n, q): RngIntElt, RngIntElt → GrpMatGL(n, K): RngIntElt, FldFin → GrpMatGL(V): ModTupRng → GrpMatSpecialLinearGroup(n, q): RngIntElt, RngIntElt → GrpMatSpecialLinearGroup(n, K): RngIntElt, FldFin → GrpMatSpecialLinearGroup(V): ModTupRng → GrpMatSL(n, q): RngIntElt, RngIntElt → GrpMatSL(n, K): RngIntElt, FldFin → GrpMatSL(V): ModTupRng → GrpMatAffineGeneralLinearGroup(GrpMat, n, q): Cat, RngIntElt, RngIntElt → GrpMatAffineGeneralLinearGroup(GrpMat, n, K): Cat, RngIntElt, FldFin → GrpMatAffineGeneralLinearGroup(GrpMat, V): Cat, ModTupRng → GrpMatAffineGeneralLinearGroup(E): GrpPerm → GrpPermAGL(GrpMat, n, q): Cat, RngIntElt, RngIntElt → GrpMatAGL(GrpMat, n, K): Cat, RngIntElt, FldFin → GrpMatAGL(GrpMat, V): Cat, ModTupRng → GrpMatAGL(E): GrpPerm → GrpPermAffineSpecialLinearGroup(GrpMat, n, q): Cat, RngIntElt, RngIntElt → GrpMatAffineSpecialLinearGroup(GrpMat, n, K): Cat, RngIntElt, FldFin → GrpMatAffineSpecialLinearGroup(GrpMat, V): Cat, ModTupRng → GrpMatASL(GrpMat, n, q): Cat, RngIntElt, RngIntElt → GrpMatASL(GrpMat, n, K): Cat, RngIntElt, FldFin → GrpMatASL(GrpMat, V): Cat, ModTupRng → GrpMat
- Unitary Groups
ConformalUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpMatConformalUnitaryGroup(n, K): RngIntElt, FldFin → GrpMatConformalUnitaryGroup(V): ModTupRng → GrpMatCU(n, q): RngIntElt, RngIntElt → GrpMatCU(n, K): RngIntElt, FldFin → GrpMatCU(V): ModTupRng → GrpMatGeneralUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpMatGeneralUnitaryGroup(n, K): RngIntElt, FldFin → GrpMatGeneralUnitaryGroup(V): ModTupRng → GrpMatGU(n, q): RngIntElt, RngIntElt → GrpMatGU(n, K): RngIntElt, FldFin → GrpMatGU(V): ModTupRng → GrpMatSpecialUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpMatSpecialUnitaryGroup(n, K): RngIntElt, FldFin → GrpMatSpecialUnitaryGroup(V): ModTupRng → GrpMatSU(n, q): RngIntElt, RngIntElt → GrpMatSU(n, K): RngIntElt, FldFin → GrpMatSU(V): ModTupRng → GrpMatConformalSpecialUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpMatCSU(n, q): RngIntElt, RngIntElt → GrpMat
- Symplectic Groups
ConformalSymplecticGroup(n, q): RngIntElt, RngIntElt → GrpMatConformalSymplecticGroup(n, K): RngIntElt, FldFin → GrpMatConformalSymplecticGroup(V): ModTupRng → GrpMatCSp(n, q): RngIntElt, RngIntElt → GrpMatCSp(n, K): RngIntElt, FldFin → GrpMatCSp(V): ModTupRng → GrpMatSymplecticGroup(n, q): RngIntElt, RngIntElt → GrpMatSymplecticGroup(n, K): RngIntElt, FldFin → GrpMatSymplecticGroup(V): ModTupRng → GrpMatSp(n, q): RngIntElt, RngIntElt → GrpMatSp(n, K): RngIntElt, FldFin → GrpMatSp(V): ModTupRng → GrpMat
- Orthogonal and Spin Groups
ConformalOrthogonalGroup(n, q): RngIntElt, RngIntElt → GrpMatConformalOrthogonalGroup(n, K): RngIntElt, FldFin → GrpMatConformalOrthogonalGroup(V): ModTupRng → GrpMatCO(n, q): RngIntElt, RngIntElt → GrpMatCO(n, K): RngIntElt, FldFin → GrpMatCO(V): ModTupRng → GrpMatGeneralOrthogonalGroup(n, q): RngIntElt, RngIntElt → GrpMatGeneralOrthogonalGroup(n, K): RngIntElt, FldFin → GrpMatGeneralOrthogonalGroup(V): ModTupRng → GrpMatGO(n, q): RngIntElt, RngIntElt → GrpMatGO(n, K): RngIntElt, FldFin → GrpMatGO(V): ModTupRng → GrpMat- Example: Orthogonal Group in Odd Dimension
SpecialOrthogonalGroup(n, q): RngIntElt, RngIntElt → GrpMatSpecialOrthogonalGroup(n, K): RngIntElt, FldFin → GrpMatSpecialOrthogonalGroup(V): ModTupRng → GrpMatSO(n, q): RngIntElt, RngIntElt → GrpMatSO(n, K): RngIntElt, FldFin → GrpMatSO(V): ModTupRng → GrpMatConformalSpecialOrthogonalGroup(n, q): RngIntElt, RngIntElt → GrpMatCSO(n, q): RngIntElt, RngIntElt → GrpMatConformalOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt → GrpMatConformalOrthogonalGroupPlus(n, K): RngIntElt, FldFin → GrpMatConformalOrthogonalGroupPlus(V): ModTupRng → GrpMatCOPlus(n, q): RngIntElt, RngIntElt → GrpMatCOPlus(n, K): RngIntElt, FldFin → GrpMatCOPlus(V): ModTupRng → GrpMatGeneralOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt → GrpMatGeneralOrthogonalGroupPlus(n, K): RngIntElt, FldFin → GrpMatGeneralOrthogonalGroupPlus(V): ModTupRng → GrpMatGOPlus(n, q): RngIntElt, RngIntElt → GrpMatGOPlus(n, K): RngIntElt, FldFin → GrpMatGOPlus(V): ModTupRng → GrpMatSpecialOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt → GrpMatSpecialOrthogonalGroupPlus(n, K): RngIntElt, FldFin → GrpMatSpecialOrthogonalGroupPlus(V): ModTupRng → GrpMatSOPlus(n, q): RngIntElt, RngIntElt → GrpMatSOPlus(n, K): RngIntElt, FldFin → GrpMatSOPlus(V): ModTupRng → GrpMatConformalSpecialOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt → GrpMatCSOPlus(n, q): RngIntElt, RngIntElt → GrpMatConformalOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt → GrpMatConformalOrthogonalGroupMinus(n, K): RngIntElt, FldFin → GrpMatConformalOrthogonalGroupMinus(V): ModTupRng → GrpMatCOMinus(n, q): RngIntElt, RngIntElt → GrpMatCOMinus(n, K): RngIntElt, FldFin → GrpMatCOMinus(V): ModTupRng → GrpMatGeneralOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt → GrpMatGeneralOrthogonalGroupMinus(n, K): RngIntElt, FldFin → GrpMatGeneralOrthogonalGroupMinus(V): ModTupRng → GrpMatGOMinus(n, q): RngIntElt, RngIntElt → GrpMatGOMinus(n, K): RngIntElt, FldFin → GrpMatGOMinus(V): ModTupRng → GrpMatSpecialOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt → GrpMatSpecialOrthogonalGroupMinus(n, K): RngIntElt, FldFin → GrpMatSpecialOrthogonalGroupMinus(V): ModTupRng → GrpMatSOMinus(n, q): RngIntElt, RngIntElt → GrpMatSOMinus(n, K): RngIntElt, FldFin → GrpMatSOMinus(V): ModTupRng → GrpMatConformalSpecialOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt → GrpMatCSOMinus(n, q): RngIntElt, RngIntElt → GrpMatOmega(n, q): RngIntElt, RngIntElt → GrpMatOmega(n, K): RngIntElt, FldFin → GrpMatOmega(V): ModTupRng → GrpMatOmegaPlus(n, q): RngIntElt, RngIntElt → GrpMatOmegaPlus(n, K): RngIntElt, FldFin → GrpMatOmegaPlus(V): ModTupRng → GrpMatOmegaMinus(n, q): RngIntElt, RngIntElt → GrpMatOmegaMinus(n, K): RngIntElt, FldFin → GrpMatOmegaMinus(V): ModTupRng → GrpMatSpin(n, q): RngIntElt, RngIntElt → GrpMatSpin(n, K): RngIntElt, FldFin → GrpMatSpin(V): ModTupRng → GrpMatSpinPlus(n, q): RngIntElt, RngIntElt → GrpMatSpinPlus(n, K): RngIntElt, FldFin → GrpMatSpinPlus(V): ModTupRng → GrpMatSpinMinus(n, q): RngIntElt, RngIntElt → GrpMatSpinMinus(n, K): RngIntElt, FldFin → GrpMatSpinMinus(V): ModTupRng → GrpMat
- Exceptional Groups
- Group Recognition
- Constructive Recognition of Alternating Groups
RecogniseAlternatingOrSymmetric(G : parameters): Grp, RngIntElt → BoolElt, Map, Map, Map, Map, BoolEltAlternatingOrSymmetricElementToWord(G, g): Grp, GrpElt → BoolElt, GrpSLPElt- Example: Recognising a Representation of A13
RecogniseSymmetric(G, n: parameters): Grp, RngIntElt → BoolElt, Map, Map, Map, Map, BoolEltSymmetricElementToWord(G, g): Grp, GrpElt → BoolElt, GrpSLPEltRecogniseAlternating(G, n: parameters): Grp, RngIntElt → BoolElt, Map, Map, Map, Map, BoolEltAlternatingElementToWord(G, g): Grp, GrpElt → BoolElt, GrpSLPEltGuessAltsymDegree(G: parameters): Grp → BoolElt, MonStgElt, RngIntElt- Example: Guessing the Degree of an Alternating Group
- Determining the Type of a Finite Group of Lie Type
- Classical Forms
ClassicalForms(G: parameters): GrpMat → RecSymplecticForm(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: Classical Forms of an Orthogonal Group
TransformForm(form, type): AlgMatElt, MonStgElt → GrpMatEltTransformForm(G): GrpMat → GrpMatEltSpinorNorm(g, form): GrpMatElt, AlgMatElt → RngIntElt- Example: Spinor Norm in an Orthogonal Group
- Recognizing Classical Groups in their Natural Representation
- Constructive Recognition of Linear Groups
RecognizeSL2(G): GrpMat → BoolElt, Map, Map, Map, MapRecogniseSL2(G): GrpMat → BoolElt, Map, Map, Map, MapRecognizeSL2(G): GrpPerm → BoolElt, Map, Map, Map, MapRecogniseSL2(G): GrpPerm → BoolElt, Map, Map, Map, MapRecognizeSL2(G, q): GrpMat, RngIntElt → BoolElt, Map, Map, Map, MapRecogniseSL2(G, q): GrpMat, RngIntElt → BoolElt, Map, Map, Map, MapRecognizeSL2(G, q): GrpPerm, RngIntElt → BoolElt, Map, Map, Map, MapRecogniseSL2(G, q): GrpPerm, RngIntElt → BoolElt, Map, Map, Map, MapSL2ElementToWord(G, g): GrpMat, GrpMatElt → BoolElt, GrpSLPEltSL2ElementToWord(G, g): GrpPerm, GrpPermElt → BoolElt, GrpSLPEltSL2Characteristic(G : parameters): GrpMat → RngIntElt, RngIntEltSL2Characteristic(G : parameters): GrpPerm → RngIntElt, RngIntElt- Example: Recognising SL2 in Its Natural Representation
- Example: Recognising SL2 in Dimension 6
RecogniseSL3(G): GrpMat → BoolElt, Map, Map, Map, MapRecogniseSL3(G, q : parameters): GrpMat, RngIntElt → BoolElt, Map, Map, Map, MapSL3ElementToWord(G, g): GrpMat, GrpMatElt → BoolElt, GrpSLPElt- Example: Recognising SL3 and Its Symmetric Square
RecogniseSL(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, MapRecognizeSL(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, Map
- Constructive Recognition of Symplectic Groups
RecogniseSpOdd(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, MapRecognizeSpOdd(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, MapRecogniseSp4(G, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, Map, Map, Map, SeqEnum, SeqEnumRecognizeSp4(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, MapRecognizeSU3(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, MapRecogniseSU4(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, MapRecognizeSU4(G, d, q): Grp, RngIntElt, RngIntElt → BoolElt, Map, Map
- Recognition Of Classical Groups in Low Degree
RecogniseSmallDegree(G): GrpMat → BoolElt, GrpMatRecogniseSmallDegree(G, type, d, q): GrpMat, MonStgElt, RngIntElt, RngIntElt → BoolElt, GrpMat, Map, MapSmallDegreePreimage(G, g): GrpMat, GrpMatElt → GrpMatEltSmallDegreeImage(G, h): GrpMat, GrpMatElt → GrpMatElt- Example: Recognising SL4 in a Symmetric Square
- Constructive Recognition of Suzuki Groups
- Introduction
- Recognition Functions
IsSuzukiGroup(G): GrpMat → BoolElt, RngIntEltRecogniseSz(G : parameters): GrpMat → BoolElt, Map, Map, Map, MapRecognizeSz(G : parameters): GrpMat → BoolElt, Map, Map, Map, MapSzElementToWord(G, g): GrpMat, GrpMatElt → BoolElt, GrpSLPEltSzPresentation(q): RngIntElt → GrpFP, HomGrpSatisfiesSzPresentation(G): GrpMat → BoolEltSuzukiIrreducibleRepresentation(F, twists : parameters): FldFin, SeqEnum[RngIntElt] → GrpMat- Example: Recognising the Suzuki Group Sz(32)
- Example: Constructive Membership in 2.Sz(8)
- Example: Suzuki Group in Dimension 64
- Example: Suzuki Group in Cross Characteristic
- Constructive Recognition of Small Ree Groups
- Introduction
- Recognition Functions
RecogniseRee(G : parameters): GrpMat → BoolElt, Map, Map, Map, MapRecognizeRee(G : parameters): GrpMat → BoolElt, Map, Map, Map, MapReeElementToWord(G, g): GrpMat, GrpMatElt → BoolElt, GrpSLPEltIsReeGroup(G): GrpMat → BoolElt, RngIntEltReeIrreducibleRepresentation(F, twists : parameters): FldFin, SeqEnum[RngIntElt] → GrpMat- Example: Recognising a Ree Group
- Constructive Recognition of Large Ree Groups
- Introduction
- Recognition Functions
RecogniseLargeRee(G : parameters): GrpMat → BoolElt, Map, Map, Map, MapRecognizeLargeRee(G : parameters): GrpMat → BoolElt, Map, Map, Map, MapLargeReeElementToWord(G, g): GrpMat, GrpMatElt → BoolElt, GrpSLPEltIsLargeReeGroup(G): GrpMat → BoolElt, RngIntElt
- Properties of Finite Groups Of Lie Type
- Maximal Subgroups of the Classical Groups
- Maximal Subgroups of the Exceptional Groups
SuzukiMaximalSubgroups(G): GrpMat → SeqEnum, SeqEnumSuzukiMaximalSubgroupsConjugacy(G, R, S): GrpMat, GrpMat, GrpMat → GrpMatElt, GrpSLPEltReeMaximalSubgroups(G): GrpMat → SeqEnum, SeqEnumReeMaximalSubgroupsConjugacy(G, R, S): GrpMat, GrpMat, GrpMat → GrpMatElt, GrpSLPEltSzMaximals(q): RngIntElt → SeqEnumReeMaximals(q): RngIntElt → SeqEnumG2Maximals(q): RngIntElt → SeqEnum
- Sylow Subgroups of the Classical Groups
ClassicalSylow(G,p): GrpMat, RngIntElt → GrpMatClassicalSylowConjugation(G,P,S): GrpMat, GrpMat, GrpMat → GrpMatEltClassicalSylowNormaliser(G,P): GrpMat, GrpMat → GrpMatEltClassicalSylowToPC(G,P): GrpMat, GrpMat → GrpPC, UserProgram, Map- Example: Sylow Subgroups of a Symplectic Group
- Sylow Subgroups of Exceptional Groups
SuzukiSylow(G, p): GrpMat, RngIntElt → GrpMat, SeqEnumSuzukiSylowConjugacy(G, R, S, p): GrpMat, GrpMat, GrpMat, RngIntElt → GrpMatElt, GrpSLPElt- Example: Sylow Subgroups of a Large Suzuki Group
ReeSylow(G, p): GrpMat, RngIntElt → GrpMat, SeqEnumReeSylowConjugacy(G, R, S, p): GrpMat, GrpMat, GrpMat, RngIntElt → GrpMatElt, GrpSLPEltLargeReeSylow(G, p): GrpMat, RngIntElt → GrpMat, SeqEnum- Example: Sylow Subgroups of a Ree Group
- 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, SeqEnumStandardGeneratorsGroupNames() → SetIndxStandardCopy(str): MonStgElt → Grp, BoolEltIsomorphismToStandardCopy(G, str : parameters): Grp, MonStgElt → BoolElt, MapStandardPresentation(G, str : parameters): Grp, MonStgElt → BoolElt, SeqEnum, SeqEnumMaximalSubgroups(G, str : parameters): Grp, MonStgElt → BoolElt, SeqEnum, SeqEnumSubgroups(G, str : parameters): Grp, MonStgElt → BoolElt, SeqEnumGoodBasePoints(G, str : parameters): Grp, MonStgElt → BoolElt, SeqEnumSubgroupsData(str): MonStgElt → SeqEnumMaximalSubgroupsData(str : parameters): MonStgElt → SeqEnum- Example: Atlas Data for the Janko Group J1
- Automorphism Groups of Finite Simple Groups
- Bibliography