Groups

• Introduction
  • The Categories of Finite Groups
• Construction of Elements
  • Construction of an Element
    • elt< G | L >: Grp, List(Elt) → GrpElt
    • G ! Q: Grp, [ Elt ] → GrpElt
    • Identity(G): Grp → GrpElt
    • Id(G): Grp → GrpElt
  • Coercion
    • G ! g: Grp, GrpElt → GrpElt
  • Homomorphisms
    • hom< G -> H | L >: Grp, Grp → Map
    • hom< G -> H | x :-> e(x) >: Grp, Grp → Map
    • IdentityHomomorphism(G): Grp → Map
    • Example: Homomorphisms
    • Example: Homomorphisms 2
  • Arithmetic with Elements
    • g * h: GrpElt, GrpElt → GrpElt
    • g ^ n: GrpElt, RngIntElt → GrpElt
    • g / h: GrpElt, GrpElt → GrpElt
    • g ^ h: GrpElt, GrpElt → GrpElt
    • (g, h): GrpElt, GrpElt → GrpElt
    • (g₁, ..., gᵣ): GrpElt, ..., GrpElt → GrpElt
    • g eq h: GrpElt, GrpElt → BoolElt
    • g ne h: GrpElt, GrpElt → BoolElt
    • IsId(g): GrpElt → BoolElt
    • IsIdentity(g): GrpElt → BoolElt
    • Order(g): GrpElt → RngIntElt
    • Example: Arithmetic
• Construction of a General Group
  • The General Group Constructors
    • PermutationGroup< X | L >: Set, List → GrpPerm, Hom
    • PermutationGroup< n | L >: RngIntElt, List → GrpPerm, Hom
    • MatrixGroup< n, R | L >: RngIntElt, Rng, List → GrpMat, Hom
    • Group< X | R >: List(Identifiers), List(GrpFPRel) → GrpFP, Hom(Grp)
    • PolycyclicGroup< X | R >: List(Identifiers), List(GrpFPRel) → GrpPC, Hom
    • AbelianGroup< X | R >: List(Identifiers), List(GrpAbRel) → GrpAb, Hom(GrpAb)
    • Example: Group Constructors
    • Example: Polycyclic Group
  • Construction of Subgroups
    • sub<G | L>: Grp, List → Grp
    • ncl<G | L>: Grp, List → Grp
    • Example: Subgroup
  • Construction of Quotient Groups
    • quo<G | L>: Grp, List → Grp, Map
    • G / N: Grp, Grp → Grp
    • Example: Quotient
• Standard Groups and Extensions
  • Construction of a Standard Group
    • AbelianGroup(C, Q): Cat, [ RngIntElt ] → GrpFin
    • AbelianGroup(Q): [ RngIntElt ] → GrpAb
    • AlternatingGroup(C, n): Cat, RngIntElt → GrpFin
    • AlternatingGroup(n): RngIntElt → GrpPerm
    • Alt(C, n): Cat, RngIntElt → GrpFin
    • Alt(n): RngIntElt → GrpPerm
    • CyclicGroup(C, n): Cat, RngIntElt → GrpFin
    • CyclicGroup(n): RngIntElt → GrpPerm
    • DihedralGroup(C, n): Cat, RngIntElt → GrpFin
    • DihedralGroup(n): RngIntElt → GrpPerm
    • DicyclicGroup(n): RngIntElt → GrpFP
    • DicyclicGroup(A, a): GrpAb, GrpAbElt → GrpFP
    • SymmetricGroup(C, n): Cat, RngIntElt → GrpFin
    • SymmetricGroup(n): RngIntElt → GrpPerm
    • Sym(GrpFin, n): Cat, RngIntElt → GrpFin
    • Sym(n): RngIntElt → GrpPerm
    • ExtraSpecialGroup(C, p, n : parameters): Cat, RngIntElt, RngIntElt → GrpFin
    • ExtraSpecialGroup(p, n : parameters): RngIntElt, RngIntElt → GrpPerm
    • Example: Standard Groups
  • Construction of Extensions
    • DirectProduct(G, H): Grp, Grp → Grp
    • DirectProduct(Q): [ Grp ] → Grp
    • SemidirectProduct(K, H, f: parameters): Grp, Grp, Map → Grp, Map, Map, Map
    • Example: semidirect
    • AffineSplitExtension(M: parameters): ModGrp → Grp, Map, Map, Map
    • Example: Affine Split
    • Example: Extensions
• Transfer Functions Between Group Categories
  • pQuotient(F, p, c: parameters): GrpFP, RngIntElt, RngIntElt → GrpPC, Map
  • CosetAction(G, H): Grp, Grp → Hom(Grp), GrpPerm, Grp
  • RegularRepresentation(G, H): Grp, Grp → Hom(Grp), GrpPerm, Grp
  • CosetImage(G, H): Grp, Grp → GrpPerm
  • CosetKernel(G, H): Grp, Grp → Grp
  • MinimalDegreePermutationRepresentation(G: parameters): Grp → Hom(Grp), GrpPerm
  • Example: Minimal Degree Permutation Representation
  • PermutationRepresentationQuotient(G, N : parameters): Grp, Grp → Hom(Grp), GrpPerm
  • GPCGroup(G): Grp → GrpGPC, Hom(Grp)
  • PCGroup(G): Grp → GrpPC, Hom(Grp)
  • FPGroup(G: parameters): GrpPerm → GrpFP, Hom(Grp)
  • Example: Coset Action
  • Example: CosetAction 2
  • Example: FP Group
• Basic Operations
  • Accessing Group Information
    • G . i: Grp, RngIntElt → GrpElt
    • Generators(G): Grp → { GrpFinElt }
    • NumberOfGenerators(G): Grp → RngIntElt
    • Ngens(G): Grp → RngIntElt
    • SmallestGeneratingSet(G: parameters): Grp → SetIndx
    • Generic(G): Grp → Grp
    • Parent(g): GrpElt → Grp
    • Example: Generators
    • Orbit(G, M, x): Grp, Any, Any → {Any}
    • OrbitClosure(G, M, S): Grp, Any, {Any} → Any
  • Names of Finite Groups
    • GroupName(G): Grp → MonStgElt
    • Example: Grp Groupname
    • Group(s): MonStgElt → Grp
    • Example: Grp Group
• Operations on the Set of Elements
  • Order and Index Functions
    • Order(G): GrpFin → RngIntElt
    • # G: GrpFin → RngIntElt
    • FactoredOrder(G): GrpFin → [ <RngIntElt, RngIntElt> ]
    • Index(G, H): GrpFin, GrpFin → RngIntElt
    • FactoredIndex(G, H): GrpFin, GrpFin → [ <RngIntElt, RngIntElt> ]
    • Example: Order
  • Membership and Equality
    • g in G: GrpFinElt, GrpFin → BoolElt
    • g notin G: GrpFinElt, GrpFin → BoolElt
    • S subset G: { GrpFinElt }, GrpFin → BoolElt
    • S notsubset G: { GrpFinElt }, GrpFin → BoolElt
    • H subset G: GrpFin, GrpFin → BoolElt
    • IsSubgroup(H,G): GrpFin, GrpFin → BoolElt
    • H notsubset G: GrpFin, GrpFin → BoolElt
    • H eq G: GrpFin, GrpFin → BoolElt
    • H ne G: GrpFin, GrpFin → BoolElt
  • Set Operations
    • NumberingMap(G): GrpFin → Map
    • Representative(G): GrpFin → GrpFinElt
    • Rep(G): GrpFin → GrpFinElt
    • Example: Set Operations
  • Random Elements
    • Random(G: parameters): GrpFin → GrpFinElt
    • Example: Random Operations
    • RandomProcess(G): GrpFin → Process
    • RandomProcessWithWords(G): GrpFin → Process
    • RandomProcessWithValues(G, Q): GrpFin, SeqEnum → Process
    • RandomProcessWithWordsAndValues(G, Q): GrpFin, SeqEnum → Process
    • Random(P): Process → GrpFinElt
    • Random(P): Process → GrpFinElt
    • InitialiseProspector(G:parameters): GrpMat
    • InitialiseProspector(G:parameters): GrpPerm
    • Prospector(G, f:parameters): Grp, UserProgram → BoolElt, GrpElt, GrpSLPElt
    • Example: Random Prospector
  • Action on a Coset Space
    • CosetTable(G, H): GrpFin, GrpFin → Map
    • CosetTable(G, f): GrpFin, Hom(GrpFin) → Hom(GrpFin)
    • Transversal(G, H): Grp, Grp → { @ GrpElt  @}, Map
    • RightTransversal(G, H): Grp, Grp → { @ GrpElt  @}, Map
    • CosetAction(G, H): Grp, Grp → Hom(Grp), GrpPerm, Grp
    • CosetImage(G, H): Grp, Grp → GrpPerm
    • CosetKernel(G, H): Grp, Grp → Grp
• Standard Subgroup Constructions
  • H ^ g: GrpFin, GrpFinElt → GrpFin
  • Conjugate(H, g): GrpFin, GrpFinElt → GrpFin
  • H meet K: GrpFin, GrpFin → GrpFin
  • CommutatorSubgroup(G, H, K): GrpFin, GrpFin, GrpFin → GrpFin
  • CommutatorSubgroup(H, K): GrpFin, GrpFin → GrpFin
  • Centralizer(G, g): GrpFin, GrpFinElt → GrpFin
  • Centraliser(G, g): GrpFin, GrpFinElt → GrpFin
  • Centralizer(G, H): GrpFin, GrpFin → GrpFin
  • Centraliser(G, H): GrpFin, GrpFin → GrpFin
  • Core(G, H): GrpFin, GrpFin → GrpFin
  • H ^ G: GrpFin, GrpFin → GrpFin
  • NormalClosure(G, H): GrpFin, GrpFin → GrpFin
  • Normalizer(G, H): GrpFin, GrpFin → GrpFin
  • Normaliser(G, H): GrpFin, GrpFin → GrpFin
  • pCore(G, p): GrpFin, RngIntElt → GrpFin
  • SylowSubgroup(G, p): GrpFin, RngIntElt → GrpFin
  • Sylow(G, p): GrpFin, RngIntElt → GrpFin
  • Abstract Group Predicates
    • IsAbelian(G): GrpFin → BoolElt
    • IsCyclic(G): GrpFin → BoolElt
    • IsElementaryAbelian(G): GrpFin → BoolElt
    • IsCentral(G, H): GrpFin, GrpFin → BoolElt
    • IsConjugate(G, g, h): GrpFin, GrpFinElt, GrpFinElt → BoolElt, GrpFinElt
    • IsConjugate(G, H, K): GrpFin, GrpFin, GrpFin → BoolElt, GrpFinElt
    • IsExtraSpecial(G): GrpFin → BoolElt
    • IsHyperelementary(G): Grp → BoolElt, Grp, Grp
    • Example: Grp Ishyperelementary
    • IsMaximal(G, H): GrpFin, GrpFin → BoolElt
    • IsNilpotent(G): GrpFin → BoolElt
    • IsNormal(G, H): GrpFin, GrpFin → BoolElt
    • IsPerfect(G): GrpFin → BoolElt
    • IsQGroup(G): Grp → BoolElt
    • Example: Grp Isqgroup
    • IsSelfNormalizing(G, H): GrpFin, GrpFin → BoolElt
    • IsSelfNormalising(G, H): GrpFin, GrpFin → BoolElt
    • IsSimple(G): GrpFin → BoolElt
    • IsSoluble(G): GrpFin → BoolElt
    • IsSolvable(G): GrpFin → BoolElt
    • IsSpecial(G): GrpFin → BoolElt
    • IsAbstractFrobeniusGroup(G): GrpFin → BoolElt, Grp, Grp
    • IsSubnormal(G, H): GrpFin, GrpFin → BoolElt
    • IsTrivial(G): Grp → BoolElt
• Characteristic Subgroups and Normal Structure
  • Characteristic Subgroups and Subgroup Series
    • Centre(G): GrpFin → GrpFin
    • Center(G): GrpFin → GrpFin
    • Hypercentre(G): GrpFin → GrpFin
    • Hypercenter(G): GrpFin → GrpFin
    • DerivedLength(G): GrpFin → RngIntElt
    • DerivedSeries(G): GrpFin → [ GrpFin ]
    • DerivedSubgroup(G): GrpFin → GrpFin
    • DerivedGroup(G): GrpFin → GrpFin
    • FittingSubgroup(G): GrpFin → GrpFin
    • FrattiniSubgroup(G): GrpFin → GrpFin
    • JenningsSeries(G): GrpFin → [ GrpFin ]
    • LowerCentralSeries(G): GrpFin → [ GrpFin ]
    • NilpotencyClass(G): GrpFin → RngIntElt
    • H ^ G: GrpFin → GrpFin
    • NormalClosure(G, H): GrpFin, GrpFin → GrpFin
    • NormalLattice(G): GrpFin → NormalLattice
    • NormalSubgroups(G): GrpFin → [ Rec ]
    • pCentralSeries(G, p): GrpFin, RngIntElt → [ GrpFin ]
    • Radical(G): GrpFin → GrpFin
    • SolubleResidual(G): GrpFin → GrpFin
    • SolvableResidual(G): GrpFin → GrpFin
    • SubnormalSeries(G, H): GrpFin, GrpFin → [ GrpFin ]
    • UpperCentralSeries(G): GrpFin → [ GrpFin ]
  • The Abstract Structure of a Group
    • CompositionFactors(G): GrpFin → [ <RngIntElt, RngIntElt, RngIntElt> ]
    • PrimaryAbelianInvariants(G): GrpFin → [ RngIntElt ]
    • AbelianInvariants(G): GrpFin → [ RngIntElt ]
    • PrimaryAbelianBasis(G): GrpFin → [ GrpFinElt ], [ RngIntElt ]
    • AbelianBasis(G): GrpFin → [ GrpFinElt ], [ RngIntElt ]
• Conjugacy Classes of Elements
  • Class(H, x): GrpFin, GrpFinElt → { GrpFinElt }
  • Conjugates(H, x): GrpFin, GrpFinElt → { GrpFinElt }
  • ClassMap(G: parameters): GrpFin → Map
  • ConjugacyClasses(G: parameters): GrpFin → [ <RngIntElt, RngIntElt, GrpFinElt> ]
  • Classes(G: parameters): GrpFin → [ <RngIntElt, RngIntElt, GrpFinElt> ]
  • ClassesData(G: parameters): GrpFin → [ <RngIntElt, RngIntElt> ]
  • ClassRepresentative(G, x): GrpFin, GrpFinElt → GrpFinElt
  • IsConjugate(G, g, h): GrpFin, GrpFinElt, GrpFinElt → BoolElt, GrpFinElt
  • IsConjugate(G, H, K): GrpFin, GrpFin, GrpFin → BoolElt, GrpFinElt
  • Exponent(G): GrpFin → RngIntElt
  • NumberOfClasses(G): GrpFin → RngIntElt
  • Nclasses(G): GrpFin → RngIntElt
  • PowerMap(G): GrpFin → Map
  • Example: Classes
• Conjugacy Classes of Subgroups
  • Conjugacy Classes of Subgroups
    • SubgroupClasses(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • Subgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • Subgroups(G, N: parameters): GrpFin, GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • ElementaryAbelianSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • AbelianSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • CyclicSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • NilpotentSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • SolubleSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • SolvableSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • NonsolvableSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • PerfectSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • SimpleSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • RegularSubgroups(G: parameters): GrpFin → [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
    • SetVerbose("SubgroupLattice", i): MonStgElt, RngIntElt
    • Class(G, H): GrpFin, GrpFin → { GrpFin }
    • Conjugates(G, H): GrpFin, GrpElt → { GrpElt }
    • Example: Subgroups
  • The Poset of Subgroup Classes
    • Creating the Poset of Subgroup Classes
      • SubgroupLattice(G): GrpFin → SubGrpLat
      • Example: Create Subgroup Poset
    • Operations on Subgroup Class Posets
      • # L: SubGrpLat → RngIntElt
      • L ! i: SubGrpLat, RngIntElt → SubGrpLatElt
      • L ! H: SubGrpLat, GrpFin → SubGrpLatElt
      • Bottom(L): SubGrpLat → SubGrpLatElt
      • Top(L): SubGrpLat → SubGrpLatElt
      • Random(L): SubGrpLat → SubGrpLatElt
      • Example: Lattice Operations
    • Operations on Poset Elements
      • IntegerRing() ! e: SubGrpLatElt → RngIntElt
      • e eq f: SubGrpLatElt, SubGrpLatElt → SubGrpLatElt
      • e ge f: SubGrpLatElt, SubGrpLatElt → BoolElt
      • e ge f: SubGrpLatElt, SubGrpLatElt → BoolElt
      • e le f: SubGrpLatElt, SubGrpLatElt → BoolElt
      • e subset f: SubGrpLatElt, SubGrpLatElt → SubGrpLatElt
      • e lt f: SubGrpLatElt, SubGrpLatElt → BoolElt
    • Class Information from a Conjugacy Class Poset
      • Group(e): SubGrpLatElt → GrpFin
      • Centraliser(e, f): SubGrpLatElt, SubGrpLatElt → SubGrpLatElt
      • Centralizer(e, f): SubGrpLatElt, SubGrpLatElt → SubGrpLatElt
      • Normaliser(e, f): SubGrpLatElt, SubGrpLatElt → SubGrpLatElt
      • Normalizer(e, f): SubGrpLatElt, SubGrpLatElt → SubGrpLatElt
      • Length(e): SubGrpLatElt → RngIntElt
      • Order(e): SubGrpLatElt → RngIntElt
      • MaximalSubgroups(e): SubGrpLatElt → { SubGrpLatElt }
      • MinimalOvergroups(e): SubGrpLatElt → { SubGrpLatElt }
      • NumberOfInclusions(e, f): SubGrpLatElt, SubGrpLatElt → RngIntElt
• All Subgroups and Intermediate Subgroups
  • AllSubgroups(G): GrpFin → [ SeqEnum ]
  • MinimalOvergroups(G,H): GrpFin, GrpFin → [ SeqEnum ]
  • IntermediateSubgroups(G,H: parameters): GrpFin, GrpFin → [ SeqEnum ]
• Cohomology
  • pMultiplicator(G, p): GrpFin, RngIntElt → [ RngIntElt ]
  • pCover(G, F, p): GrpPerm, GrpFP, RngIntElt → GrpFinFP
  • CohomologicalDimension(G, M, i): GrpFin, ModRng, RngIntElt → RngIntElt
  • ExtensionProcess(G, M, F): GrpPerm, ModRng, GrpFP → GrpFPExtProc
  • Extension(P, Q): Process → GrpFinFP
  • NextExtension(P): Process → GrpFinFP
  • SplitExtension(G, M, F): GrpPerm, ModRng, GrpFP → GrpFP
• Characters and Representations
  • Character Theory
    • CharacterDegrees(G): GrpPC → [ Tup ]
    • CharacterTable(G): GrpFin → TabChtr
    • PermutationCharacter(G): GrpPerm → AlgChtrElt
    • PermutationCharacter(G, H): GrpFin, GrpFin → AlgChtrElt
    • BurnsideCokernel(G): Grp → GrpAb, UserProgram, SeqEnum[AlgChtrElt]
    • Example: Grp Burnsidecokernel
  • Representation Theory
    • GModule(G, S): GrpFin, AlgMat → ModGrpFin
    • GModule(G, A, B): GrpFin, GrpFin, GrpFin → ModGrpFin, Map
    • PermutationModule(G, H, R): GrpFin, GrpFin, Rng → ModGrpFin
    • PermutationModule(G, R): GrpPerm, Rng → ModGrpFin
    • Example: Modules
    • Example: Modules 2
• Databases of Groups