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PERMUTATION GROUPS

 
Acknowledgements
 
Introduction
      Terminology
      The Category of Permutation Groups
      The Construction of a Permutation Group
 
Creation of a Permutation Group
      Construction of the Symmetric Group
      Construction of a Permutation
      Construction of a General Permutation Group
 
Elementary Properties of a Group
      Accessing Group Information
      Group Order
      Abstract Properties of a Group
 
Homomorphisms
 
Building Permutation Groups
      Some Standard Permutation Groups
      Direct Products and Wreath Products
 
Permutations
      Coercion
      Arithmetic with Permutations
      Properties of Permutations
      Predicates for Permutations
      Set Operations
 
Conjugacy
 
Subgroups
      Construction of a Subgroup
      Membership and Equality
      Elementary Properties of a Subgroup
      Standard Subgroups
      Maximal Subgroups
      Conjugacy Classes of Subgroups
      Classes of Subgroups Satisfying a Condition
 
Quotient Groups
      Construction of Quotient Groups
      Abelian, Nilpotent and Soluble Quotients
 
Permutation Group Actions
      G-Sets
      Creating a G-Set
      Images, Orbits and Stabilizers
      Action on a G-Space
      Action on Orbits
      Action on a G-invariant Partition
      Action on a Coset Space
      Reduced Permutation Actions
      The Jellyfish Algorithm
 
Normal and Subnormal Subgroups
      Characteristic Subgroups and Normal Series
      Maximal and Minimal Normal Subgroups
      Lattice of Normal Subgroups
      Composition and Chief Series
      The Socle
      The Soluble Radical and its Quotient
      Complements and Supplements
      Abelian Normal Subgroups
 
Cosets and Transversals
      Cosets
      Transversals
 
Presentations
      Generators and Relations
      Permutations as Words
 
Automorphism Groups
 
Cohomology
 
Representation Theory
 
Identification
      Identification as an Abstract Group
      Identification as a Permutation Group
 
Base and Strong Generating Set
      Construction of a Base and Strong Generating Set
      Defining Values for Attributes
      Accessing the Base and Strong Generating Set
      Working with a Base and Strong Generating Set
      Modifying a Base and Strong Generating Set
 
Permutation Representations of Linear Groups
 
Permutation Group Databases
 
Ordered Partition Stacks
      Construction of Ordered Partition Stacks
      Properties of Ordered Partition Stacks
      Operations on Ordered Partition Stacks
 
Bibliography







DETAILS

 
Introduction

      Terminology

      The Category of Permutation Groups

      The Construction of a Permutation Group

 
Creation of a Permutation Group

      Construction of the Symmetric Group
            Sym(n) : RngIntElt -> GrpPerm
            Sym(X) : Set -> GrpPerm
            StandardGroup(G) : GrpPerm -> GrpPerm, Map
            Example GrpPerm_Sym (H58E1)

      Construction of a Permutation
            elt< G | L > : GrpPerm, List(Elt) -> GrpPermElt
            G ! Q : GrpPerm, [ Elt ] -> GrpPermElt
            G ! (...)(...)...(...) : GrpPerm, Cycles -> GrpPermElt
            G ! \(...)(...)...(...) : GrpPerm, LiteralCycles -> GrpPermElt
            G ! Q : GrpPerm, SeqEnum[SetIndx] -> GrpPermElt
            ElementToSequence(g) : GrpPermElt -> [ Elt ]
            Identity(G) : Grp -> GrpPermElt
            Example GrpPerm_Permutations (H58E2)

      Construction of a General Permutation Group
            PermutationGroup< X | L > : Set, List -> GrpPerm
            PermutationGroup< n | L > : RngIntElt, List -> GrpPerm
            Example GrpPerm_Hessian (H58E3)

 
Elementary Properties of a Group

      Accessing Group Information
            G . i : GrpPerm, RngIntElt -> GrpPermElt
            Degree(G) : GrpPermElt -> RngIntElt
            Generators(G) : GrpPerm -> { GrpPermElt }
            GeneratorsSequence(G) : GrpPerm -> [ GrpPermElt ]
            NumberOfGenerators(G) : GrpPerm -> RngIntElt
            FewGenerators(G) : GrpPerm -> [GrpPermElt]
            Generic(G) : GrpPerm -> GrpPerm
            Parent(g) : GrpPermElt -> GrpPerm
            GSet(G) : GrpPerm -> GSet
            Example GrpPerm_BasicAccess (H58E4)

      Group Order
            Order(G) : GrpPerm -> RngIntElt
            FactoredOrder(G) : GrpPerm -> [ <RngIntElt, RngIntElt> ]

      Abstract Properties of a Group
            IsAbelian(G) : GrpPerm -> BoolElt
            IsCyclic(G) : GrpPerm -> BoolElt
            IsElementaryAbelian(G) : GrpPerm -> BoolElt
            IsSpecial(G) : GrpPerm -> BoolElt
            IsExtraSpecial(G) : GrpPerm -> BoolElt
            IsNilpotent(G) : GrpPerm -> BoolElt
            IsSoluble(G) : GrpPerm -> BoolElt
            IsPerfect(G) : GrpPerm -> BoolElt
            IsSimple(G) : GrpPerm -> BoolElt
            IsWreathProduct(G) : GrpPerm -> BoolElt, GrpPerm, GrpPerm, GrpPerm
            Example GrpPerm_BasicProperties (H58E5)

 
Homomorphisms
      hom<G | L> : GrpPerm, List -> Map
      Domain(f) : Map -> Grp
      Codomain(f) : Map -> Grp
      Image(f) : Map -> Grp
      Kernel(f) : Map -> Grp
      IsHomomorphism(G, H, Q) : GrpPerm, GrpPerm, SeqEnum[GrpPermElt] -> Bool, Map
      Example GrpPerm_Homomorphism (H58E6)

 
Building Permutation Groups

      Some Standard Permutation Groups
            AbelianGroup(GrpPerm, Q) : Cat, [ RngIntElt ] -> GrpPerm
            AlternatingGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
            CyclicGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
            DihedralGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
            Sym(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
            ExtraSpecialGroup(GrpPerm, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpPerm
            YoungSubgroup(L) : [RngIntElt] -> GrpPerm
            Example GrpPerm_StandardGroups (H58E7)

      Direct Products and Wreath Products
            DirectProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]
            DirectProduct(Q) : [ GrpPerm ] -> GrpPerm, [ Hom(Grp) ], [ Hom(Grp) ]
            PrimitiveWreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm
            PrimitiveWreathProduct(Q) : [ GrpPerm ] -> GrpPerm
            WreathProduct(G, H) : GrpPerm, GrpPerm -> GrpPerm, SeqEnum[Map], Map, Map
            WreathProduct(Q) : [ GrpPerm ] -> GrpPerm
            WreathProduct(B) : GSet -> GrpPerm, GrpPerm, GrpPerm
            WreathProduct(G, B) : GrpPerm, GSet -> GrpPerm, GrpPerm, GrpPerm
            Example GrpPerm_Products (H58E8)

 
Permutations

      Coercion
            G ! g : GrpPerm, GrpPermElt -> GrpPermElt
            G !! H : GrpPerm, GrpPerm -> GrpPerm

      Arithmetic with Permutations
            g * h : GrpPermElt, GrpPermElt -> GrpPermElt
            g ^ n : GrpPermElt, RngIntElt -> GrpPermElt
            g / h : GrpPermElt, GrpPermElt -> GrpPermElt
            g ^ h : GrpPermElt, GrpPermElt -> GrpPermElt
            (g, h) : GrpPermElt, GrpPermElt -> GrpPermElt
            (g1, ..., gr) : GrpPermElt, ..., GrpPermElt -> GrpPermElt

      Properties of Permutations
            CycleStructure(g) : GrpPermElt -> [ <RngIntElt, RngIntElt> ]
            Degree(g) : GrpPermElt -> RngIntElt
            IsEven(g) : GrpPermElt -> BoolElt
            Sign(g) : GrpPermElt -> RngIntElt
            Order(g) : GrpPermElt -> RngIntElt

      Predicates for Permutations
            g eq h : GrpPermElt, GrpPermElt -> BoolElt
            g ne h : GrpPermElt, GrpPermElt -> BoolElt
            IsId(g) : GrpPermElt -> BoolElt
            Example GrpPerm_Arithmetic (H58E9)

      Set Operations
            G * H : GrpPerm, GrpPerm -> { GrpPermElt }
            ElementSet(G, H) : GrpPerm, GrpPerm -> { GrpPermElt }
            NumberingMap(G) : GrpPerm -> Map
            RandomProcess(G) : GrpPerm -> Process
            Random(G: parameters) : GrpPerm -> GrpPermElt
            Random(P) : Process -> GrpPermElt
            Representative(G) : GrpPerm -> GrpPermElt
            Example GrpPerm_SetOperations (H58E10)
            Example GrpPerm_SetOperations-2 (H58E11)

 
Conjugacy
      Class(H, x) : GrpPerm, GrpPermElt -> { GrpPermElt }
      ConjugacyClasses(G: parameters) : GrpPerm -> [ <RngIntElt, RngIntElt, GrpPermElt> ]
      ClassRepresentative(G, x) : GrpPerm, GrpPermElt -> GrpPermElt
      ClassCentraliser(G, i) : GrpPerm, RngIntElt -> GrpPerm
      ClassMap(G: parameters) : GrpPerm -> Map
      IsConjugate(G, g, h: parameters) : GrpPerm, GrpPermElt, GrpPermElt -> BoolElt, GrpPermElt
      IsConjugate(G, H, K: parameters) : GrpPerm, GrpPerm, GrpPerm -> BoolElt, GrpPermElt
      Exponent(G) : GrpPerm -> RngIntElt
      NumberOfClasses(G) : GrpPerm -> RngIntElt
      PowerMap(G) : GrpPerm -> Map
      AssertAttribute(G, "Classes", Q) : GrpPerm, MonStgElt, SeqEnum ->
      Example GrpPerm_Classes (H58E12)
      Example GrpPerm_Classes-2 (H58E13)

 
Subgroups

      Construction of a Subgroup
            sub<G | L> : GrpPerm, List -> GrpPerm
            ncl<G | L> : GrpPerm, List -> GrpPerm
            Example GrpPerm_Constructors (H58E14)
            Example GrpPerm_Constructors-2 (H58E15)
            Example GrpPerm_Constructors-3 (H58E16)

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

      Elementary Properties of a Subgroup
            Index(G, H) : GrpPerm, GrpPerm -> RngIntElt
            FactoredIndex(G, H) : GrpPerm, GrpPerm -> [ <RngIntElt, RngIntElt> ]
            IsCentral(G, H) : GrpPerm -> BoolElt
            IsNormal(G, H) : GrpPerm, GrpPerm -> BoolElt
            IsSelfNormalizing(G, H) : GrpPerm, GrpPerm -> BoolElt
            IsSubnormal(G, H) : GrpPerm, GrpPerm -> BoolElt

      Standard Subgroups
            H ^ g : GrpPerm, GrpPermElt -> GrpPerm
            H meet K : GrpPerm, GrpPerm -> GrpPerm
            IntersectionWithNormalSubgroup(G, N: parameters) : GrpPerm, GrpPerm -> GrpPerm
            CommutatorSubgroup(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> GrpPerm
            Centralizer(G, g: parameters) : GrpPerm, GrpPermElt -> GrpPerm
            Centralizer(G, H) : GrpPerm, GrpPerm -> GrpPerm
            CentralizerOfNormalSubgroup(G, H) : GrpPerm, GrpPerm -> GrpPerm
            SectionCentraliser(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> GrpPerm
            Core(G, H) : GrpPerm, GrpPerm -> GrpPerm
            H ^ G : GrpPerm, GrpPerm -> GrpPerm
            Normalizer(G, H: parameters) : GrpPerm, GrpPerm -> GrpPerm
            SymmetricNormalizer(G) : GrpPerm -> GrpPerm
            SylowSubgroup(G, p) : GrpPerm, RngIntElt -> GrpPerm
            Example GrpPerm_SubgroupConstructions (H58E17)

      Maximal Subgroups
            IsMaximal(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt
            IsProbablyMaximal(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt
            MaximalSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            Example GrpPerm_Maximals (H58E18)

      Conjugacy Classes of Subgroups
            SubgroupClasses(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            SubgroupsLift(G, A, B, Q: parameters) : GrpPerm, GrpPerm, GrpPerm, SeqEnum -> SeqEnum
            LowIndexSubgroups(G, n: parameters) : GrpPerm, RngIntElt -> SeqEnum
            Example GrpPerm_Subgroups (H58E19)
            Example GrpPerm_Subgroups-2 (H58E20)

      Classes of Subgroups Satisfying a Condition
            NormalSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            ElementaryAbelianSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            CyclicSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            AbelianSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            NilpotentSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            SolvableSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            PerfectSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            NonsolvableSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
            SimpleSubgroups(G: parameters) : GrpPerm -> [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]

 
Quotient Groups

      Construction of Quotient Groups
            quo<G | L> : GrpPerm, List -> GrpPerm, Map
            G / N : GrpPerm, GrpPerm -> GrpPerm
            Example GrpPerm_Quotient (H58E21)

      Abelian, Nilpotent and Soluble Quotients
            AbelianQuotient(G) : GrpPerm -> GrpAb, Map
            ElementaryAbelianQuotient(G, p) : GrpPerm, RngIntElt -> GrpAb, Map
            pQuotient(G, p, c) : GrpPerm, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt
            NilpotentQuotient(G, c) : GrpPerm, RngIntElt -> GrpGPC, Map
            SolvableQuotient(G): GrpPerm, RngIntElt -> GrpPC, Map, SeqEnum, MonStgElt
            Example GrpPerm_SpecialQuotient (H58E22)

 
Permutation Group Actions

      G-Sets

      Creating a G-Set
            GSetFromIndexed(G, Y) : GrpPerm, SetIndx -> GSet
            GSet(G, X, Y) : GrpPerm, GSet, SetEnum -> GSet
            GSet(G) : GrpPerm -> GSet
            GSet(G, Y, f) : GrpPerm, Set, Map -> GSet
            Action(Y) : GSet -> Map
            Group(Y) : GSet -> GrpPerm
            Labelling(G) : GrpPerm -> SetIndx
            Degree(g, Y) : GrpPermElt, GSet -> RngIntElt
            Degree(G, Y) : GrpPerm, GSet -> RngIntElt
            Support(g, Y) : GrpPermElt, GSet -> { Elt }
            Support(G, Y) : GrpPerm, GSet -> { Elt }
            Example GrpPerm_GSets (H58E23)

      Images, Orbits and Stabilizers
            x ^ g : Elt, GrpPermElt -> Elt
            Image(g, Y, y) : GrpPermElt, GSet, Elt -> Elt
            Fix(g, Y): GrpPermElt, GSet -> { Elt }
            Fix(G, Y) : GrpPerm, GSet -> { Elt }
            x ^ G : Elt, GrpPerm -> GSet
            Cycle(e, x) : GrpPermElt, Elt -> SetIndx
            CycleDecomposition(e) : GrpPermElt -> SeqEnum[SetIndx]
            Orbit(G, Y, y) : GrpPerm, GSet, Elt -> GSet
            Orbits(G, Y) : GrpPerm, GSet -> [ GSet ]
            OrbitRepresentatives(G) : GrpPerm -> SeqEnum
            OrbitClosure(G, Y, S) : GrpPerm, GSet, { Elt } -> GSet
            IsConjugate(G, Y, y, z) : GrpPerm, GSet, Elt, Elt -> BoolElt, GrpPermElt
            Stabilizer(G, Y, y) : GrpPerm, GSet, Elt -> GrpPerm
            IsPrimitive(G, Y) : GrpPerm, GSet -> BoolElt
            IsTransitive(G, Y) : GrpPerm, GSet -> BoolElt
            IsTransitive(G, Y, k) : GrpPerm, GSet, RngIntElt -> BoolElt
            IsSharplyTransitive(G, Y, k) : GrpPerm, GSet, RngIntElt -> BoolElt
            Transitivity(G, Y) : GrpPerm, GSet -> RngIntElt
            IsRegular(G, Y) : GrpPerm, GSet -> BoolElt
            IsSemiregular(G, Y) : GrpPerm, GSet -> BoolElt
            IsSemiregular(G, Y, S) : GrpPerm, GSet, SetEnum -> BoolElt
            IsFrobenius(G) : GrpPerm -> BoolElt
            Example GrpPerm_Stabilizers (H58E24)

      Action on a G-Space
            Action(G, Y) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
            ActionImage(G, Y) : GrpPerm, GSet -> GrpPerm
            ActionKernel(G, Y) : GrpPerm, GSet -> GrpPerm
            IsFaithful(G, Y) : : GrpPerm, GSet -> BoolElt
            Example GrpPerm_Actions (H58E25)

      Action on Orbits
            OrbitAction(G, T) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
            OrbitImage(G, T) : GrpPerm, GSet -> GrpPerm
            OrbitKernel(G, T) : GrpPerm, GSet -> GrpPerm
            IsOrbit(G, S) : GrpPerm, { Elt } -> BoolElt
            Example GrpPerm_OrbitActions (H58E26)

      Action on a G-invariant Partition
            IsBlock(G, S) : GrpPerm, { Elt } -> BoolElt
            IsPrimitive(G) : GrpPerm -> BoolElt
            MaximalPartition(G) : GrpPerm -> GSet
            MinimalPartition(G: parameters) : GrpPerm -> GSet
            MinimalPartitions(G: parameters) : GrpPerm -> [ GSet ]
            AllPartitions(G) : GrpPerm -> SetEnum
            BlocksAction(G, P) : GrpPerm, Any -> Hom(GrpPerm), GrpPerm, GrpPerm
            BlocksImage(G, P) : GrpPerm, Any -> GrpPerm
            BlocksKernel(G, P) : GrpPerm, Any -> GrpPerm
            Example GrpPerm_BlocksActions (H58E27)
            Example GrpPerm_BlocksActions-2 (H58E28)

      Action on a Coset Space
            CosetAction(G, H: parameters) : Grp, Grp -> Hom(Grp), GrpPerm, GrpPerm
            CosetImage(G, H: parameters) : Grp, Grp -> GrpPerm
            CosetKernel(G, H) : Grp, Grp -> Grp

      Reduced Permutation Actions
            TransitiveQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm
            PrimitiveQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm
            DegreeReduction(G) : GrpPerm -> GrpPerm, Hom

      The Jellyfish Algorithm
            JellyfishConstruction(G: parameters) : GrpPerm -> BoolElt
            JellyfishImage(G) : GrpPerm -> GrpPerm
            JellyfishImage(G, x) : GrpPerm, GrpPermElt -> GrpPermElt
            JellyfishPreimage(G, x) : GrpPerm, GrpPermElt -> GrpPermElt

 
Normal and Subnormal Subgroups

      Characteristic Subgroups and Normal Series
            DerivedSeries(G) : GrpPerm -> [ GrpPerm ]
            CompositionSeries(G) : GrpPerm -> [ GrpPerm ]
            CommutatorSubgroup(G) : GrpPerm -> GrpPerm
            SolubleResidual(G) : GrpPerm -> GrpPerm
            DerivedLength(G) : GrpPerm -> RngIntElt
            LowerCentralSeries(G) : GrpPerm -> [ GrpPerm ]
            NilpotencyClass(G) : GrpPerm -> RngIntElt
            UpperCentralSeries(G) : GrpPerm -> [ GrpPerm ]
            Centre(G) : GrpPerm -> GrpPerm
            Hypercentre(G) : GrpPerm -> GrpPerm
            pCore(G, p) : GrpPerm, RngIntElt -> GrpPerm
            pCoreQuotient(G, p) : GrpPerm, RngIntElt -> GrpPerm, Map, GrpPerm
            FittingSubgroup(G) : GrpPerm -> GrpPerm
            FrattiniSubgroup(G) : GrpPerm -> GrpPerm
            JenningsSeries(G) : GrpPerm -> [ GrpPerm ]
            pCentralSeries(G, p) : GrpPerm, RngIntElt -> [ GrpPerm ]
            SubnormalSeries(G, H) : GrpPerm, GrpPerm -> [ GrpPerm ]
            Example GrpPerm_Series (H58E29)

      Maximal and Minimal Normal Subgroups
            MaximalNormalSubgroup(G) : GrpPerm -> GrpPerm
            MinimalNormalSubgroups(G) : GrpPerm -> [ GrpPerm ]

      Lattice of Normal Subgroups
            NormalSubgroups(G) : GrpPerm -> [ Rec ]
            NormalLattice(G) : GrpPerm -> SubGrpLat
            Example GrpPerm_NormalSubgroups (H58E30)

      Composition and Chief Series
            ChiefFactors(G) : GrpPerm -> [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
            ChiefSeries(G) : GrpPerm -> [ GrpPerm ], [ <RngIntElt, RngIntElt, RngIntElt, RngIntElt> ]
            CompositionFactors(G) : GrpPerm -> [ <RngIntElt, RngIntElt, RngIntElt> ]
            Example GrpPerm_CompFactors (H58E31)

      The Socle
            Socle(G) : GrpPerm -> GrpPerm
            SocleFactor(G) : GrpPerm -> GrpPerm
            SocleFactors(G) : GrpPerm -> [ GrpPerm ]
            SocleSeries(G) : GrpPerm -> [ GrpPerm ]
            EARNS(G) : GrpPerm -> GrpPerm
            IsAffine(G) : GrpPerm -> BoolElt, GrpPerm
            AffineAction(G) : GrpPerm -> Hom, GrpPerm, GrpPerm
            AffineImage(G) : GrpPerm -> GrpPerm
            AffineKernel(G) : GrpPerm -> GrpPerm
            SocleAction(G) : GrpPerm -> Hom, GrpPerm, GrpPerm
            SocleImage(G) : GrpPerm -> GrpPerm
            SocleKernel(G) : GrpPerm -> GrpPerm
            SocleQuotient(G) : GrpPerm -> GrpPerm, Hom, GrpPerm
            RefineSection(G, M, N) : GrpPerm, GrpPerm, GrpPerm -> [ GrpPerm ]
            Example GrpPerm_PrimitiveStructure (H58E32)

      The Soluble Radical and its Quotient
            Radical(G) : GrpPerm -> GrpPerm
            RadicalQuotient(G) : GrpPerm -> GrpPerm, Hom(GrpPerm), GrpPerm
            ElementaryAbelianSeries(G: parameters) : GrpPerm -> [ GrpPerm ]
            ElementaryAbelianSeriesCanonical(G) : GrpPerm -> [ GrpPerm ]
            Example GrpPerm_Radical (H58E33)

      Complements and Supplements
            Complements(G, M) : GrpPerm, GrpPerm -> [ GrpPerm ]
            Complements(G, M, N) : GrpPerm, GrpPerm, GrpPerm -> [ GrpPerm ]
            HasComplement(G, M) : GrpPerm, GrpPerm -> BoolElt, GrpPerm
            Supplements(G, M) : GrpPerm, GrpPerm -> [ GrpPerm ]
            Supplements(G, M, N) : GrpPerm, GrpPerm, GrpPerm -> [ GrpPerm ]
            HasSupplement(G, M) : GrpPerm, GrpPerm -> BoolElt, GrpPerm
            Example GrpPerm_Complements (H58E34)

      Abelian Normal Subgroups
            AbelianNormalSubgroup(G) : GrpPerm -> GrpPerm
            AbelianNormalQuotient(G, H) : GrpPerm -> GrpPerm, Hom(GrpPerm), GrpPerm
            SolubleNormalQuotient(G, H) : GrpPerm -> GrpPerm, Hom(GrpPerm), GrpPerm
            ElementaryAbelianNormalSubgroup(G) : GrpPerm -> GrpPerm
            pElementaryAbelianNormalSubgroup(G, p) : GrpPerm, RngIntElt -> GrpPerm
            MEANS(G) : GrpPerm -> GrpPerm
            MEANS(G, N) : GrpPerm, GrpPerm -> GrpPerm

 
Cosets and Transversals

      Cosets
            H * g : GrpPerm, GrpPermElt -> Elt
            DoubleCoset(G, H, g, K) : GrpPerm, GrpPerm, GrpPermElt, GrpPerm -> GrpPermDcosElt
            DoubleCosetRepresentatives(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> SeqEnum, SeqEnum
            ProcessLadder(L, G, U) : [GrpPerm], GrpPerm, GrpPerm -> Rec
            GetRep(p, R) : GrpPermElt, Rec -> GrpPermElt
            DeleteData(R) : Rec ->
            YoungSubgroupLadder(L) : [RngIntElt] -> [GrpPerm]
            StabilizerLadder(G, d) : GrpPerm, RngMPolElt -> [GrpPerm]
            x in C : GrpPermElt, Elt -> BoolElt
            x notin C : GrpPermElt, Elt -> BoolElt
            C1 eq C2 : Elt, Elt -> BoolElt
            C1 ne C2 : Elt, Elt -> BoolElt
            # C : Elt -> RngIntElt
            CosetTable(G, H) : Grp, Grp -> Map
            [Future release] CosetTable(G, f) : Grp, Map -> Map

      Transversals
            Transversal(G, H) : GrpPerm, GrpPerm -> {@ GrpPermElt atbrace, Map
            TransversalProcess(G, H) : GrpPerm, GrpPerm -> GrpPermTransProc
            TransversalProcessRemaining(P) : GrpPermTransProc -> RngIntElt
            TransversalProcessNext(P) : GrpPermTransProc -> GrpPermElt
            ShortCosets(p, H, G) : GrpPermElt, GrpPerm, GrpPerm -> [GrpPermElt]

 
Presentations

      Generators and Relations
            FPGroup(G) : GrpPerm :-> GrpFP, Hom(Grp)
            FPQuotient(G, N) : GrpPerm, GrpPerm :-> GrpFP, Hom(Grp)
            FPGroupStrong(G: parameters) : GrpPerm :-> GrpFP, Hom(Grp)

      Permutations as Words
            WordGroup(G) : GrpPerm -> GrpBB, Map
            InverseWordMap(G) : GrpPerm -> Map
            ActingWord(G, x, y) : GrpPerm, Elt, Elt -> GrpFPElt

 
Automorphism Groups
      AutomorphismGroup(G: parameters) : GrpPerm -> GrpAuto
      IsIsomorphic(G, H: parameters) : GrpPerm, GrpPerm -> BoolElt, Hom(Grp)
      Example GrpPerm_Automorphisms (H58E35)

 
Cohomology
      pMultiplicator(G, p) : GrpPerm, RngIntElt -> [ RngIntElt ]
      pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFP
      CohomologicalDimension(G, M, i) : GrpPerm, ModRng, RngIntElt -> RngIntElt
      ExtensionProcess(G, M, F) : GrpPerm, ModRng, GrpFP -> Process
      Extension(P, Q) : Process -> GrpFP
      SplitExtension(G, M, F) : GrpPerm, ModRng, GrpFP -> GrpFP
      Example GrpPerm_Cohomology (H58E36)
      Example GrpPerm_Cohomology-2 (H58E37)

 
Representation Theory
      CharacterTable(G: parameters) : GrpPerm -> TabChtr
      PermutationCharacter(G) : GrpPerm -> AlgChtrElt
      PermutationCharacter(G, H) : GrpPerm, GrpPerm -> AlgChtrElt
      GModule(G, S) : Grp, AlgMat -> ModGrp
      GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
      PermutationModule(G, H, R) : Grp, Grp, Rng -> ModGrp
      PermutationModule(G, R) : GrpPerm, Rng -> ModGrp
      Example GrpPerm_GModule (H58E38)

 
Identification

      Identification as an Abstract Group
            NameSimple(G) : GrpPerm -> <RngIntElt, RngIntElt, RngIntElt>

      Identification as a Permutation Group
            IsAlternating(G) : GrpPerm -> BoolElt
            IsSymmetric(G) : GrpPerm -> BoolElt
            IsAltsym(G) : GrpPerm -> BoolElt
            TwoTransitiveGroupIdentification(G) : GrpPerm -> Tup
            RecogniseAlternatingOrSymmetric(G, n) : Grp, RngIntElt -> BoolElt, BoolElt, UserProgram, UserProgram
            IsEven(G): GrpPerm -> BoolElt
            Example GrpPerm_RecogniseAltsym1 (H58E39)
            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 GrpPerm_RecogniseAltsym2 (H58E40)

 
Base and Strong Generating Set

      Construction of a Base and Strong Generating Set
            BSGS(G) : GrpPerm ->
            SimsSchreier(G: parameters) : GrpPerm : ->
            RandomSchreier(G: parameters) : GrpPerm : ->
            ToddCoxeterSchreier(G: parameters) : GrpPerm : ->
            SolubleSchreier(G: parameters) : GrpPerm : ->
            Verify(G: parameters ) : RngIntElt ->
            Example GrpPerm_BSGS (H58E41)
            Example GrpPerm_BSFS-2 (H58E42)

      Defining Values for Attributes
            AssertAttribute(G, "Order", n) : GrpPerm, MonStgElt, RngIntElt ->
            AssertAttribute(G, "Order", Q) : GrpPerm, MonStgElt, [<RngIntElt, RngIntElt>] ->
            [Future release] AssertAttribute(G, "BSGS", S) : GrpPerm, MonStgElt, GrpPermBSGS ->
            Example GrpPerm_RandomSchreier (H58E43)

      Accessing the Base and Strong Generating Set
            Base(G) : GrpPerm -> [Elt]
            BasePoint(G, i) : GrpPerm, RngIntElt -> Elt
            BasicOrbit(G, i) : GrpPerm, RngIntElt -> SetIndx
            BasicOrbits(G) : GrpPerm -> [SetIndx]
            BasicOrbitLength(G, i) : GrpPerm, RngIntElt -> RngIntElt
            BasicOrbitLengths(G) : GrpPerm -> [RngIntElt]
            BasicStabilizer(G, i) : GrpPerm, RngIntElt -> GrpPerm
            BasicStabilizerChain(G) : GrpPerm -> [GrpPerm]
            IsMemberBasicOrbit(G, i, a) : GrpPerm, RngIntElt, Elt -> BoolElt
            NumberOfStrongGenerators(G) : GrpPerm -> RngIntElt
            NumberOfStrongGenerators(G, i) : GrpPerm, RngIntElt -> RngIntElt
            SchreierVectors(G) : GrpPerm -> [ [RngIntElt] ]
            SchreierVector(G, i) : GrpPerm, RngIntElt -> [RngIntElt]
            StrongGenerators(G) : GrpPerm -> SetIndx(GrpPermElt)
            StrongGenerators(G, i) : GrpPerm, RngIntElt -> SetIndx(GrpPermElt)

      Working with a Base and Strong Generating Set
            BaseImage(x) : GrpPermElt -> [Elt]
            Permutation(G, Q) : GrpPerm, [Elt] -> GrpPermElt
            SVPermutation(G, i, a) : GrpPerm, RngIntElt, Elt -> GrpPermElt
            SVWord(G, i, a) : GrpPerm, RngIntElt, Elt -> GrpFPElt
            Strip(H, x) : GrpPerm, GrpPermElt -> BoolElt, GrpPermElt, RngIntElt
            WordStrip(H, x) : GrpPerm, GrpPermElt -> BoolElt, GrpFPElt, RngIntElt
            BaseImageWordStrip(H, x) : GrpPerm, GrpPermElt -> BoolElt, GrpFPElt, RngIntElt
            WordInStrongGenerators(H, x) : GrpPerm, GrpPermElt -> GrpFPElt

      Modifying a Base and Strong Generating Set
            ChangeBase(~G, Q) : GrpPerm, [Elt] ->
            AddNormalizingGenerator(~H, x) : GrpPerm, GrpPermElt ->
            ReduceGenerators(~G) : GrpPerm ->

 
Permutation Representations of Linear Groups
      AffineGeneralLinearGroup(arguments)
      AffineSpecialLinearGroup(arguments)
      AffineGammaLinearGroup(arguments)
      AffineSigmaLinearGroup(arguments)
      ProjectiveGeneralLinearGroup(arguments)
      ProjectiveSpecialLinearGroup(arguments)
      ProjectiveGammaLinearGroup(arguments)
      ProjectiveSigmaLinearGroup(arguments)
      ProjectiveGeneralUnitaryGroup(arguments)
      ProjectiveSpecialUnitaryGroup(arguments)
      ProjectiveGammaUnitaryGroup(arguments)
      ProjectiveSigmaUnitaryGroup(arguments)
      ProjectiveSymplecticGroup(arguments)
      ProjectiveSigmaSymplecticGroup(arguments)
      PGO(arguments)
      PGOPlus(arguments)
      PGOMinus(arguments)
      PSO(arguments)
      PSOPlus(arguments)
      PSOMinus(arguments)
      ProjectiveOmega(arguments)
      ProjectiveOmegaPlus(arguments)
      ProjectiveOmegaMinus(arguments)
      ProjectiveSuzukiGroup(arguments)
      AffineGroup(M) : GrpMat[FldFin] -> GrpPerm, { at ModTupFldElt atbrace

 
Permutation Group Databases

 
Ordered Partition Stacks

      Construction of Ordered Partition Stacks
            OrderedPartitionStack(n) : RngIntElt -> StkPtnOrd
            OrderedPartitionStackZero(n, h) : RngIntElt, RngIntElt -> StkPtnOrd

      Properties of Ordered Partition Stacks
            Degree(P) : StkPtnOrd -> RngIntElt
            Height(P) : StkPtnOrd -> RngIntElt
            NumberOfCells(P, h) : StkPtnOrd, RngIntElt -> RngIntElt
            CellNumber(P, h, x) : StkPtnOrd, RngIntElt, RngIntElt -> RngIntElt
            CellSize(P, h, i) : StkPtnOrd, RngIntElt, RngIntElt -> RngIntElt
            Cell(P, h, i): StkPtnOrd, RngIntElt, RngIntElt -> SeqEnum
            Random(P, i) : StkPtnOrd, RngIntElt -> RngIntElt
            Representative(P, i) : StkPtnOrd, RngIntElt -> RngIntElt
            ParentCell(P, i) : StkPtnOrd, RngIntElt -> RngIntElt

      Operations on Ordered Partition Stacks
            SplitCell(P, i, x) : StkPtnOrd, RngIntElt, RngIntElt -> BoolElt
            SplitAllByValues(P, V) : StkPtnOrd, SeqEnum[RngIntElt] -> BoolElt, RngIntElt
            SplitCellsByValues(P, C, V) : StkPtnOrd, SeqEnum[RngIntElt], SeqEnum[RngIntElt] -> BoolElt, RngIntElt
            Pop(P) : StkPtnOrd ->
            Advance(X, L, P, h) : StkPtnOrd, seqEnum[RngIntElt], StkPtnOrd, RngIntElt ->
            Example GrpPerm_OrderedPartitionStack (H58E44)

 
Bibliography

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