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MATRIX GROUPS OVER GENERAL RINGS

 
Acknowledgements
 
Introduction
      Introduction to Matrix Groups
      The Support
      The Category of Matrix Groups
      The Construction of a Matrix Group
 
Creation of a Matrix Group
      Construction of the General Linear Group
      Construction of a Matrix Group Element
      Construction of a General Matrix Group
      Changing Rings
      Coercion between Matrix Structures
      Accessing Associated Structures
 
Homomorphisms
      Construction of Extensions
 
Operations on Matrices
      Arithmetic with Matrices
      Predicates for Matrices
      Matrix Invariants
 
Global Properties
      Group Order
      Membership and Equality
      Set Operations
 
Abstract Group Predicates
 
Conjugacy
 
Subgroups
      Construction of Subgroups
      Elementary Properties of Subgroups
      Standard Subgroups
      Low Index Subgroups
      Conjugacy Classes of Subgroups
 
Quotient Groups
      Construction of Quotient Groups
      Abelian, Nilpotent and Soluble Quotients
 
Matrix Group Actions
      Orbits and Stabilizers
      Orbit and Stabilizer Functions for Large Groups
      Action on Orbits
      Action on a Coset Space
      Action on the Natural G-Module
 
Normal and Subnormal Subgroups
      Characteristic Subgroups and Subgroup Series
      The Soluble Radical and its Quotient
      Composition and Chief Factors
 
Coset Tables and Transversals
 
Presentations
      Presentations
      Matrices as Words
 
Automorphism Groups
 
Representation Theory
 
Base and Strong Generating Set
      Introduction
      Controlling Selection of a Base
      Construction of a Base and Strong Generating Set
      Defining Values for Attributes
      Accessing the Base and Strong Generating Set
 
Soluble Matrix Groups
      Conversion to a PC-Group
      Soluble Group Functions
      p-group Functions
      Abelian Group Functions
 
Bibliography







DETAILS

 
Introduction

      Introduction to Matrix Groups

      The Support

      The Category of Matrix Groups

      The Construction of a Matrix Group

 
Creation of a Matrix Group

      Construction of the General Linear Group
            GeneralLinearGroup(n, R) : RngIntElt, Rng -> GrpMat
            Example GrpMatGen_Create (H59E1)

      Construction of a Matrix Group Element
            elt< G | L > : GrpMat, List(RngElt) -> GrpMatElt
            G ! Q : GrpMat, [ RngElt ] -> GrpMatElt
            ElementToSequence(g) : GrpMatElt -> [ RngElt ]
            Identity(G) : GrpMat -> GrpMatElt
            Example GrpMatGen_Matrices (H59E2)

      Construction of a General Matrix Group
            MatrixGroup< n, R | L > : RngIntElt, Rng, List -> GrpMat
            Example GrpMatGen_Constructor (H59E3)
            Example GrpMatGen_GLSylow (H59E4)

      Changing Rings
            ChangeRing(G, S) : GrpMat, Rng -> GrpMat, Map
            ChangeRing(G, S, f) : GrpMat, Rng, Map -> GrpMat, Map
            RestrictField(G, S) : GrpMat, FldFin -> GrpMat, Map
            ExtendField(G, L) : GrpMat, FldFin -> GrpMat, Map

      Coercion between Matrix Structures
            R ! g : AlgMat, GrpMatElt -> RngMatElt
            G ! r : GrpMat, AlgMatElt -> GrpMatElt
            M ! g : ModMatRng, GrpMatElt -> ModMatRngElt
            G ! m : GrpMat, ModMatRngElt -> GrpMatElt

      Accessing Associated Structures
            G . i : GrpMat, RngIntElt -> GrpMatElt
            Degree(G) : GrpMat -> RngIntElt
            Generators(G) : GrpMat -> { GrpMatElt }
            NumberOfGenerators(G) : GrpMat -> RngIntElt
            CoefficientRing(G) : GrpMat -> Rng
            RSpace(G) : GrpMat -> ModTupRng
            VectorSpace(G) : GrpMat -> ModTupFld
            GModule(G) : GrpMat -> ModGrp
            Generic(G) : GrpMat -> GrpMat
            Parent(G) : GrpMatElt -> GrpMat

 
Homomorphisms
      hom<G -> H | L> : GrpMat, Grp, List -> Map
      Domain(f) : Map -> Grp
      Codomain(f) : Map -> Grp
      Image(f) : Map -> Grp
      Kernel(f) : Map -> Grp
      IsHomomorphism(G, H, Q) : GrpMat, GrpMat, SeqEnum[GrpMatElt] -> Bool, Map
      Example GrpMatGen_Homomorphism (H59E5)

      Construction of Extensions
            DirectProduct(G, H) : GrpMat, GrpMat -> GrpMat
            DirectProduct(Q) : [ GrpMat ] -> GrpMat
            SemiLinearGroup(G, S) : GrpMat, FldFin -> GrpMat
            TensorWreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
            WreathProduct(G, H) : GrpMat, GrpPerm -> GrpMat
            Example GrpMatGen_Constructions (H59E6)

 
Operations on Matrices

      Arithmetic with Matrices
            g * h : GrpMatElt, GrpMatElt -> GrpMatElt
            g ^ n : GrpMatElt, RngIntElt -> GrpMatElt
            g / h : GrpMatElt, GrpMatElt -> GrpMatElt
            g ^ h : GrpMatElt, GrpMatElt -> GrpMatElt
            (g, h) : GrpMatElt, GrpMatElt -> GrpMatElt
            (g1, ..., gr) : GrpMatElt, ..., GrpMatElt -> GrpMatElt
            Example GrpMatGen_Arithmetic (H59E7)

      Predicates for Matrices
            g eq h : GrpMatElt, GrpMatElt -> BoolElt
            g ne h : GrpMatElt, GrpMatElt -> BoolElt
            IsIdentity(g) : GrpMatElt -> BoolElt
            IsScalar(g) : GrpMatElt -> BoolElt

      Matrix Invariants
            Degree(g) : GrpMatElt -> RngIntElt
            HasFiniteOrder(g) : GrpMatElt -> BoolElt, RngIntElt
            Order(g) : GrpMatElt -> RngIntElt, BoolElt
            FactoredOrder(g) : GrpMatElt -> [ <RngIntElt, RngIntElt> ], BoolElt
            ProjectiveOrder(g) : GrpMatElt -> RngIntElt, RngElt
            FactoredProjectiveOrder(A) : AlgMatElt -> [ <RngIntElt, RngIntElt> ], RngElt
            CentralOrder(g : parameters) : GrpMatElt -> RngIntElt, BoolElt
            Determinant(g) : GrpMatElt -> RngElt
            Trace(g) : GrpMatElt -> RngElt
            CharacteristicPolynomial(g: parameters) : GrpMatElt -> RngPolElt
            MinimalPolynomial(g) : GrpMatElt -> RngPolElt
            Example GrpMatGen_Invariants (H59E8)

 
Global Properties

      Group Order
            IsFinite(G) : GrpMat -> Bool, RngIntElt
            Order(G) : GrpMat -> RngIntElt
            FactoredOrder(G) : GrpMat -> [ <RngIntElt, RngIntElt> ]
            Example GrpMatGen_Order (H59E9)

      Membership and Equality
            g in G : GrpMatElt, GrpMat -> BoolElt
            g notin G : GrpMatElt, GrpMat -> BoolElt
            S subset G : { GrpMatElt }, GrpMat -> BoolElt
            H subset G : GrpMat, GrpMat -> BoolElt
            S notsubset G : { GrpMatElt }, GrpMat -> BoolElt
            H notsubset G : GrpMat, GrpMat -> BoolElt
            H eq G : GrpMat, GrpMat -> BoolElt
            H ne G : GrpMat, GrpMat -> BoolElt

      Set Operations
            NumberingMap(G) : GrpMat -> Map
            RandomProcess(G) : GrpMat -> Process
            Random(G: parameters) : GrpMat -> GrpMatElt
            Random(P) : Process -> GrpMatElt
            Example GrpMatGen_Random (H59E10)

 
Abstract Group Predicates
      IsAbelian(G) : GrpMat -> BoolElt
      IsCyclic(G) : GrpMat -> BoolElt
      IsElementaryAbelian(G) : GrpMat -> BoolElt
      IsNilpotent(G) : GrpMat -> BoolElt
      IsSoluble(G) : GrpMat -> BoolElt
      IsPerfect(G) : GrpMat -> BoolElt
      IsSimple(G) : GrpMat -> BoolElt
      Example GrpMatGen_Order (H59E11)

 
Conjugacy
      Class(H, x) : GrpMat, GrpMatElt -> { GrpMatElt }
      ClassMap(G) : GrpMat -> Map
      ConjugacyClasses(G: parameters) : GrpMat -> [ < RngIntElt, RngIntElt, GrpMatElt > ]
      ClassRepresentative(G, x) : GrpMat, GrpMatElt -> GrpMatElt
      ClassCentraliser(G, i) : GrpMat, RngIntElt -> GrpMat
      ClassInvariants(G, g) : GrpMat, GrpMatElt -> .
      ClassRepresentativeFromInvariants(G, p, h, t) : GrpMat, SeqEnum, SeqEnum, FldFinElt -> GrpMatElt
      IsConjugate(G, g, h) : GrpMat, GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt | Unass
      IsConjugate(G, H, K) : GrpMat, GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
      IsGLConjugate(H, K) : GrpMat, GrpMat -> BoolElt, GrpMatElt | Unass
      Exponent(G) : GrpMat -> RngIntElt
      NumberOfClasses(G) : GrpMat -> RngIntElt
      PowerMap(G) : GrpMat -> Map
      AssertAttribute(G, "Classes", Q) : GrpMat, MonStgElt, SeqEnum ->
      Example GrpMatGen_RationalMatrixGroupDatabase (H59E12)

 
Subgroups

      Construction of Subgroups
            sub<G | L> : GrpMat, List -> GrpMat
            ncl<G | L> : GrpMat, List -> GrpMat
            Example GrpMatGen_Subgroups (H59E13)

      Elementary Properties of Subgroups
            Index(G, H) : GrpMat, GrpMat -> RngIntElt
            FactoredIndex(G, H) : GrpMat, GrpMat -> [ <RngIntElt, RngIntElt> ]
            IsCentral(G, H) : GrpMat -> BoolElt
            IsMaximal(G, H) : GrpMat, GrpMat -> BoolElt
            IsNormal(G, H) : GrpMat, GrpMat -> BoolElt
            IsSubnormal(G, H) : GrpMat, GrpMat -> BoolElt

      Standard Subgroups
            H ^ g : GrpMat, GrpMatElt -> GrpMat
            H meet K : GrpMat, GrpMat -> GrpMat
            CommutatorSubgroup(G, H, K) : GrpMat, GrpMat, GrpMat -> GrpMat
            Centralizer(G, g) : GrpMat, GrpMatElt -> GrpMat
            Centralizer(G, H) : GrpMat, GrpMat -> GrpMat
            Core(G, H) : GrpMat, GrpMat -> GrpMat
            H ^ G : GrpMat, GrpMat -> GrpMat
            Normalizer(G, H) : GrpMat, GrpMat -> GrpMat
            SylowSubgroup(G, p) : GrpMat, RngIntElt -> GrpMat
            pCore(G, p) : GrpMat, RngIntElt -> GrpMat

      Low Index Subgroups
            LowIndexSubgroups(G, R : parameters) : GrpMat, RngIntElt -> [ GrpMat ]
            Example GrpMatGen_LowIndexMatrixGroup (H59E14)

      Conjugacy Classes of Subgroups
            SubgroupClasses(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
            MaximalSubgroups(G: parameters) : GrpMat -> [ rec< GrpMat, RngIntElt, RngIntElt, GrpFP> ]
            SubgroupsLift(G, A, B, Q: parameters) : GrpMat, GrpMat, GrpMat, SeqEnum -> SeqEnum

 
Quotient Groups

      Construction of Quotient Groups
            quo<G | L> : GrpMat, List -> GrpPerm, Map
            G / N : GrpMat, GrpMat -> GrpPerm
            Example GrpMatGen_Quotient (H59E15)

      Abelian, Nilpotent and Soluble Quotients
            AbelianQuotient(G) : GrpMat -> GrpAb, Map
            ElementaryAbelianQuotient(G, p) : GrpMat, RngIntElt -> GrpAb, Map
            pQuotient(G, p, c) : GrpMat, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt
            NilpotentQuotient(G, c) : GrpMat, RngIntElt -> GrpGPC, Map
            SolvableQuotient(G): GrpMat -> GrpPC, Map
            PCGroup(G): GrpMat -> GrpPC, Map
            Example GrpMatGen_SpecialQuotient (H59E16)

 
Matrix Group Actions

      Orbits and Stabilizers
            u * g : ModTupRngElt, GrpMatElt -> ModTupRngElt
            y ^ g : Elt, GrpMatElt -> Elt
            y ^ G : Elt, GrpMat -> SetEnum
            OrbitBounded(G, y, b) : GrpMat, Elt, RngIntElt -> BoolElt, SetEnum
            Orbits(G) : GrpMat -> [ SetIndx ]
            LineOrbits(G) : GrpMat -> [ SetIndx ]
            OrbitClosure(G, S) : GrpMat, { Elt } -> GSet
            Stabilizer(G, y) : GrpMat, Elt -> GrpMat
            Example GrpMatGen_Orbits (H59E17)

      Orbit and Stabilizer Functions for Large Groups
            OrbitsOfSpaces(G, k) : GrpMat, RngIntElt -> SeqEnum
            NumberOfFixedSpaces(x, s) : GrpMatElt, RngIntElt -> RngIntElt
            Example GrpMatGen_OrbitsOfSpaces (H59E18)
            EstimateOrbit(G, v: parameters) : GrpMat, ModTupFldElt -> RngIntElt, RngIntElt, RngIntElt
            ApproximateStabiliser(G, A, U: parameters) : GrpMat, GrpMat, ModTupFld -> GrpMat, GrpMat, RngIntElt, RngIntElt, RngIntElt
            Example GrpMatGen_OrbitsOfSpaces (H59E19)
            StabiliserOfSpaces(Q) : SeqEnum -> GrpMat, SeqEnum
            Example GrpMatGen_StabiliserOfSpaces (H59E20)
            IsUnipotent(G) : GrpMat -> BoolElt
            UnipotentStabiliser(G, U: parameters) : GrpMat, ModTupFld -> GrpMat, ModTupFld, GrpMatElt, GrpSLPElt
            Example GrpMatGen_UnipotentStabiliser (H59E21)

      Action on Orbits
            OrbitAction(G, T) : GrpMat, Elt -> Hom(Grp), GrpPerm, GrpMat
            OrbitActionBounded(G, T, b) : GrpMat, Elt, RngIntElt -> BoolElt, Hom(Grp), GrpPerm, GrpMat
            OrbitImage(G, T) : GrpMat, Set -> GrpPerm
            OrbitImageBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpPerm
            OrbitKernel(G, T) : GrpMat, Set -> GrpMat
            OrbitKernelBounded(G, T, b) : GrpMat, Set, RngIntElt -> BoolElt, GrpMat
            Example GrpMatGen_Actions (H59E22)

      Action on a Coset Space
            CosetAction(G, H) : GrpMat, GrpMat -> Hom(Grp), GrpPerm, GrpMat
            CosetImage(G, H) : GrpMat, GrpMat -> GrpPerm
            CosetKernel(G, H) : GrpMat, GrpMat -> GrpMat
            Example GrpMatGen_CosetAction (H59E23)

      Action on the Natural G-Module
            GModule(G) : GrpMat -> ModGrp
            IsIrreducible(G) : GrpMat -> BoolElt, ModGrp
            SubmoduleAction(G, S) : GrpMat -> Map, GrpMat
            SubmoduleImage(G, S) : GrpMat -> GrpMat
            QuotientModuleAction(G, S) : GrpMat -> Map, GrpMat
            QuotientModuleImage(G, S) : GrpMat -> GrpMat
            IsAbsolutelyIrreducible(G) : GrpMat -> BoolElt
            AbsoluteRepresentation(G) : GrpMat -> GrpMat, Map
            MinimalField(G) : GrpMat -> FldFin

 
Normal and Subnormal Subgroups

      Characteristic Subgroups and Subgroup Series
            Centre(G) : GrpMat -> GrpMat
            DerivedLength(G) : GrpMat -> RngIntElt
            DerivedSeries(G) : GrpMat -> [ GrpMat ]
            CommutatorSubgroup(G) : GrpMat -> GrpMat
            [Future release] FittingSubgroup(G) : GrpMat -> GrpMat
            LowerCentralSeries(G) : GrpMat -> [ GrpMat ]
            NilpotencyClass(G) : GrpMat -> RngIntElt
            H ^ G : GrpMat -> GrpMat
            SolubleResidual(G) : GrpMat -> GrpMat
            SubnormalSeries(G, H) : GrpMat, GrpMat -> [ GrpMat ]
            UpperCentralSeries(G) : GrpMat -> [ GrpMat ]
            Example GrpMatGen_Series (H59E24)

      The Soluble Radical and its Quotient
            Radical(G) : GrpMat -> GrpMat
            RadicalQuotient(G) : GrpMat -> GrpPerm, Hom(Grp), GrpMat
            ElementaryAbelianSeries(G: parameters) : GrpMat -> [ GrpMat ]
            ElementaryAbelianSeriesCanonical(G) : GrpMat -> [ GrpMat ]

      Composition and Chief Factors
            CompositionFactors(G) : : GrpMat -> [ <RngIntElt, RngIntElt, RngIntElt> ]
            ChiefFactors(G) : GrpMat -> [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
            ChiefSeries(G) : GrpMat -> [ GrpMat ], [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
            Example GrpMatGen_CompositionFactors (H59E25)

 
Coset Tables and Transversals
      CosetTable(G, H) : Grp, Grp -> Hom(Grp)
      Transversal(G, H) : GrpMat, GrpMat -> {@ GrpMatElt @}, Map

 
Presentations

      Presentations
            FPGroup(G) : GrpMat :-> GrpFP, Hom(Grp)
            FPGroupStrong(G) : GrpMat :-> GrpFP, Hom(Grp)

      Matrices as Words
            WordGroup(G) : GrpMat -> GrpSLP, Map
            InverseWordMap(G) : GrpMat -> Map

 
Automorphism Groups
      AutomorphismGroup(G: parameters) : GrpMat -> GrpAuto
      Example GrpMatGen_Automorphisms (H59E26)
      IsIsomorphic(G, H: parameters) : GrpMat, GrpMat -> BoolElt, Hom(Grp)
      Example GrpMatGen_Isomorphism (H59E27)

 
Representation Theory
      LinearCharacters(G) : GrpMat -> [ Chtr ]
      CharacterTable(G: parameters) : GrpMat -> TabChtr
      PermutationCharacter(G, H) : GrpMat, GrpMat -> AlgChtrElt
      GModule(G) : GrpMat -> ModGrp
      GModule(G, A) : GrpMat, AlgMat -> ModGrp
      GModule(G, Q) : GrpMat, [ AlgMatElt ] -> ModGrp
      GModule(G, A, B) : GrpMat, GrpMat, GrpMat -> ModGrp, Map
      PermutationModule(G, H, R) : GrpMat, GrpMat, Rng -> ModGrp
      ChangeOfBasisMatrix(G, S) : GrpMat, ModGrp -> AlgMatElt
      Example GrpMatGen_GModule (H59E28)

 
Base and Strong Generating Set

      Introduction

      Controlling Selection of a Base
            GoodBasePoints(G: parameters) : GrpMat -> []
            AssertAttribute(G, "Base", B) : GrpMat, MonStgElt, Tup ->
            HasAttribute(G, "Base") : GrpMat, MonStgElt -> BoolElt, Tup
            AssertAttribute(GrpMat, "FirstBasicOrbitBound", n) : Cat, MonStgElt, RngIntElt ->
            HasAttribute(GrpMat, "FirstBasicOrbitBound") : Cat, MonStgElt -> BoolElt, RngIntElt

      Construction of a Base and Strong Generating Set
            BSGS(G) : GrpMat ->
            RandomSchreier(G: parameters) : GrpMat ->
            ToddCoxeterSchreier(G) : GrpMat : ->
            Verify(G) : GrpMat ->

      Defining Values for Attributes
            AssertAttribute(G, "Order", n) : GrpMat, MonStgElt, RngIntElt ->
            AssertAttribute(G, "IsVerified", b) : GrpMat, MonStgElt, BoolElt ->
            HasAttribute(G, "Order") : GrpMat, MonStgElt -> RngIntElt
            HasAttribute(G, "IsVerified") : GrpMat, MonStgElt -> BoolElt

      Accessing the Base and Strong Generating Set
            Base(G) : GrpMat -> [Elt]
            BasePoint(G, i) : GrpMat, RngIntElt -> Elt
            BasicOrbit(G, i) : GrpMat, RngIntElt -> SetIndx
            BasicOrbitLength(G, i) : GrpMat, RngIntElt -> RngIntElt
            BasicOrbitLengths(G) : GrpMat -> [RngIntElt]
            BasicStabilizer(G, i) : GrpMat, RngIntElt -> GrpMat
            BasicStabilizerChain(G) : GrpMat -> [GrpMat]
            NumberOfStrongGenerators(G) : GrpMat -> RngIntElt
            StrongGenerators(G) : GrpMat -> SetIndx(GrpMat)

 
Soluble Matrix Groups

      Conversion to a PC-Group
            PolycyclicGenerators(G) : GrpMat -> [ GrpPCElt ]
            PCGroup(G) : GrpMat -> GrpPC, Map

      Soluble Group Functions
            pCentralSeries(G, p) : GrpMat, RngIntElt -> [ GrpMat ]

      p-group Functions
            IsSpecial(G) : GrpMat -> BoolElt
            IsExtraSpecial(G) : GrpMat -> BoolElt
            FrattiniSubgroup(G) : GrpMat -> GrpMat
            JenningsSeries(G) : GrpMat -> [ GrpMat ]

      Abelian Group Functions
            AbelianInvariants(G) : GrpMat -> [ RngIntElt ]

 
Bibliography

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