Permutation Groups

• 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
    • SymmetricGroup(n): RngIntElt → GrpPerm
    • Sym(X): Set → GrpPerm
    • SymmetricGroup(X): Set → GrpPerm
    • StandardGroup(G): GrpPerm → GrpPerm, Map
    • Example: Sym
  • 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 ]
    • Eltseq(g): GrpPermElt → [ Elt ]
    • Identity(G): Grp → GrpPermElt
    • Id(G): Grp → GrpPermElt
    • G ! 1: Grp, RngIntElt → GrpPermElt
    • Example: Permutations
  • Construction of a General Permutation Group
    • PermutationGroup< X | L >: Set, List → GrpPerm
    • PermutationGroup< n | L >: RngIntElt, List → GrpPerm
    • Example: Hessian
• 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
    • Ngens(G): GrpPerm → RngIntElt
    • FewGenerators(G): GrpPerm → [GrpPermElt]
    • Generic(G): GrpPerm → GrpPerm
    • Parent(g): GrpPermElt → GrpPerm
    • GSet(G): GrpPerm → GSet
    • Example: Basic Access
  • Group Order
    • Order(G): GrpPerm → RngIntElt
    • # 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
    • IsSolvable(G): GrpPerm → BoolElt
    • IsPerfect(G): GrpPerm → BoolElt
    • IsSimple(G): GrpPerm → BoolElt
    • IsWreathProduct(G): GrpPerm → BoolElt, GrpPerm, GrpPerm, GrpPerm
    • Example: Basic Properties
• Homomorphisms
  • hom<G -> H | 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: Homomorphism
• Building Permutation Groups
  • Some Standard Permutation Groups
    • AbelianGroup(GrpPerm, Q): Cat, [ RngIntElt ] → GrpPerm
    • AlternatingGroup(GrpPerm, n): Cat, RngIntElt → GrpPerm
    • AlternatingGroup(n): RngIntElt → GrpPerm
    • Alt(n): RngIntElt → GrpPerm
    • CyclicGroup(GrpPerm, n): Cat, RngIntElt → GrpPerm
    • CyclicGroup(n): RngIntElt → GrpPerm
    • DihedralGroup(GrpPerm, n): Cat, RngIntElt → GrpPerm
    • DihedralGroup(n): RngIntElt → GrpPerm
    • Sym(GrpPerm, n): Cat, RngIntElt → GrpPerm
    • SymmetricGroup(GrpPerm, n): Cat, RngIntElt → GrpPerm
    • Sym(n): RngIntElt → GrpPerm
    • SymmetricGroup(n): RngIntElt → GrpPerm
    • ExtraSpecialGroup(GrpPerm, p, n : parameters): Cat, RngIntElt, RngIntElt → GrpPerm
    • ExtraSpecialGroup(p, n : parameters): RngIntElt, RngIntElt → GrpPerm
    • YoungSubgroup(L): [RngIntElt] → GrpPerm
    • Example: Standard Groups
  • 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: Products
• 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
    • IsIdentity(g): GrpPermElt → BoolElt
    • Example: Arithmetic
  • 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
    • Rep(G): GrpPerm → GrpPermElt
    • Example: Set Operations
    • Example: SetOperations 2
• Conjugacy
  • Class(H, x): GrpPerm, GrpPermElt → { GrpPermElt }
  • Conjugates(H, x): GrpPerm, GrpPermElt → { GrpPermElt }
  • ConjugacyClasses(G: parameters): GrpPerm → [ <RngIntElt, RngIntElt, GrpPermElt> ]
  • Classes(G: parameters): GrpPerm → [ <RngIntElt, RngIntElt, GrpPermElt> ]
  • ClassRepresentative(G, x): GrpPerm, GrpPermElt → GrpPermElt
  • ClassRepresentative(G, i): GrpPerm, RngIntElt → GrpPermElt
  • ClassCentraliser(G, i): GrpPerm, RngIntElt → GrpPerm
  • ClassCentralizer(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
  • Nclasses(G): GrpPerm → RngIntElt
  • PowerMap(G): GrpPerm → Map
  • AssertAttribute(G, "Classes", Q): GrpPerm, MonStgElt, SeqEnum
  • Example: Classes
  • Example: Classes 2
• Subgroups
  • Construction of a Subgroup
    • sub<G | L>: GrpPerm, List → GrpPerm
    • ncl<G | L>: GrpPerm, List → GrpPerm
    • Example: Constructors
    • Example: Constructors 2
    • Example: Constructors 3
  • 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
    • IsSubgroup(H,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, GrpPerm → BoolElt
    • IsNormal(G, H): GrpPerm, GrpPerm → BoolElt
    • IsSelfNormalizing(G, H): GrpPerm, GrpPerm → BoolElt
    • IsSelfNormalising(G, H): GrpPerm, GrpPerm → BoolElt
    • IsSubnormal(G, H): GrpPerm, GrpPerm → BoolElt
  • Standard Subgroups
    • H ^ g: GrpPerm, GrpPermElt → GrpPerm
    • Conjugate(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
    • CommutatorSubgroup(H, K): GrpPerm, GrpPerm → GrpPerm
    • Centralizer(G, g: parameters): GrpPerm, GrpPermElt → GrpPerm
    • Centraliser(G, g: parameters): GrpPerm, GrpPermElt → GrpPerm
    • Centralizer(G, H): GrpPerm, GrpPerm → GrpPerm
    • Centraliser(G, H): GrpPerm, GrpPerm → GrpPerm
    • CentralizerOfNormalSubgroup(G, H): GrpPerm, GrpPerm → GrpPerm
    • SectionCentraliser(G, H, K): GrpPerm, GrpPerm, GrpPerm → GrpPerm
    • SectionCentralizer(G, H, K): GrpPerm, GrpPerm, GrpPerm → GrpPerm
    • Core(G, H): GrpPerm, GrpPerm → GrpPerm
    • H ^ G: GrpPerm, GrpPerm → GrpPerm
    • NormalClosure(G, H): GrpPerm, GrpPerm → GrpPerm
    • Normalizer(G, H: parameters): GrpPerm, GrpPerm → GrpPerm
    • Normaliser(G, H: parameters): GrpPerm, GrpPerm → GrpPerm
    • SymmetricNormalizer(G): GrpPerm → GrpPerm
    • SymmetricNormaliser(G): GrpPerm → GrpPerm
    • SylowSubgroup(G, p): GrpPerm, RngIntElt → GrpPerm
    • Sylow(G, p): GrpPerm, RngIntElt → GrpPerm
    • Example: Subgroup Constructions
  • 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: Maximals
    • MaximalSubgroups(G,N: parameters): GrpPerm, GrpPerm → [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
  • Conjugacy Classes of Subgroups
    • SubgroupClasses(G: parameters): GrpPerm → [ rec< GrpPerm, RngIntElt, RngIntElt, GrpFP> ]
    • Subgroups(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
    • LowIndexSubgroups(G, t: parameters): GrpPerm, Tup → SeqEnum
    • LowIndexSubgroups(G, N, n: parameters): GrpPerm, RngIntElt → SeqEnum
    • LowIndexSubgroups(G, N, t: parameters): GrpPerm, Tup → SeqEnum
    • Example: Subgroups
    • Example: Low Index Subs
    • Example: Subgroups 2
    • SubgroupLattice(G): GrpPerm → SubGrpLat
    • BurnsideMatrix(G): GrpPerm → AlgMatElt
    • DisplayBurnsideMatrix(G): GrpPerm
    • TableOfMarks(G): GrpPerm → AlgMatElt
  • 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: Quotient
  • 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 → GrpPC, Map, SeqEnum, MonStgElt
    • SolubleQuotient(G): GrpPerm → GrpPC, Map, SeqEnum, MonStgElt
    • Example: Special Quotient
• 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, Y): GrpPerm, Set → 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): GrpPermElt → RngIntElt
    • Degree(G, Y): GrpPerm, GSet → RngIntElt
    • Degree(G): GrpPerm → RngIntElt
    • Support(g, Y): GrpPermElt, GSet → { Elt }
    • Support(g): GrpPermElt → { Elt }
    • Support(G, Y): GrpPerm, GSet → { Elt }
    • Support(G): GrpPerm → { Elt }
    • Example: G Sets
  • Images, Orbits and Stabilizers
    • x ^ g: Elt, GrpPermElt → Elt
    • Image(g, Y, y): GrpPermElt, GSet, Elt → Elt
    • Image(g, y): GrpPermElt, Elt → Elt
    • Fix(g, Y): GrpPermElt, GSet → { Elt }
    • Fix(g): GrpPermElt → { Elt }
    • Fix(G, Y): GrpPerm, GSet → { Elt }
    • Fix(G): GrpPerm → { 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
    • Orbit(G, y): GrpPerm, Elt → GSet
    • Orbits(G, Y): GrpPerm, GSet → [ GSet ]
    • Orbits(G): GrpPerm → [ GSet ]
    • OrbitRepresentatives(G): GrpPerm → SeqEnum
    • OrbitClosure(G, Y, S): GrpPerm, GSet, { Elt } → GSet
    • OrbitClosure(G, S): GrpPerm, { Elt } → GSet
    • IsConjugate(G, Y, y, z): GrpPerm, GSet, Elt, Elt → BoolElt, GrpPermElt
    • IsConjugate(G, y, z): GrpPerm, Elt, Elt → BoolElt, GrpPermElt
    • Stabilizer(G, Y, y): GrpPerm, GSet, Elt → GrpPerm
    • Stabiliser(G, Y, y): GrpPerm, GSet, Elt → GrpPerm
    • Stabilizer(G, y): GrpPerm, Elt → GrpPerm
    • Stabiliser(G, y): GrpPerm, Elt → GrpPerm
    • TwoClosure(G): GrpPerm → GrpPerm
    • IsPrimitive(G, Y): GrpPerm, GSet → BoolElt
    • IsPrimitive(G): GrpPerm → BoolElt
    • IsTransitive(G, Y): GrpPerm, GSet → BoolElt
    • IsTransitive(G): GrpPerm → BoolElt
    • IsTransitive(G, Y, k): GrpPerm, GSet, RngIntElt → BoolElt
    • IsTransitive(G, k): GrpPerm, RngIntElt → BoolElt
    • IsSharplyTransitive(G, Y, k): GrpPerm, GSet, RngIntElt → BoolElt
    • IsSharplyTransitive(G, k): GrpPerm, RngIntElt → BoolElt
    • Transitivity(G, Y): GrpPerm, GSet → RngIntElt
    • Transitivity(G): GrpPerm → RngIntElt
    • Homogeneity(G): GrpPerm → RngIntElt
    • IsRegular(G, Y): GrpPerm, GSet → BoolElt
    • IsRegular(G): GrpPerm → BoolElt
    • IsSemiregular(G, Y): GrpPerm, GSet → BoolElt
    • IsSemiregular(G): GrpPerm → BoolElt
    • IsSemiregular(G, Y, S): GrpPerm, GSet, SetEnum → BoolElt
    • IsSemiregular(G, S): GrpPerm, SetEnum → BoolElt
    • IsFrobenius(G): GrpPerm → BoolElt
    • Example: Stabilizers
  • 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: Actions
  • 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: Orbit Actions
  • 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 ]
    • MinimalBlocks(G: parameters): GrpPerm → [ SetEnum ]
    • AllPartitions(G): GrpPerm → SetEnum
    • BlocksAction(G, P): GrpPerm, Any → Hom(GrpPerm), GrpPerm, GrpPerm
    • BlocksAction(G, P): GrpPerm, GSet → Hom(GrpPerm), GrpPerm, GrpPerm
    • BlocksAction(G, P): GrpPerm, SeqEnum → Hom(GrpPerm), GrpPerm, GrpPerm
    • BlocksAction(G, P): GrpPerm, SetIndx → Hom(GrpPerm), GrpPerm, GrpPerm
    • BlocksAction(G, P): GrpPerm, SetEnum → Hom(GrpPerm), GrpPerm, GrpPerm
    • BlocksImage(G, P): GrpPerm, Any → GrpPerm
    • BlocksImage(G, P): GrpPerm, GSet → GrpPerm
    • BlocksImage(G, P): GrpPerm, SetIndx → GrpPerm
    • BlocksImage(G, P): GrpPerm, SeqEnum → GrpPerm
    • BlocksImage(G, P): GrpPerm, SetEnum → GrpPerm
    • BlocksKernel(G, P): GrpPerm, Any → GrpPerm
    • BlocksKernel(G, P): GrpPerm, GSet → GrpPerm
    • BlocksKernel(G, P): GrpPerm, SetIndx → GrpPerm
    • BlocksKernel(G, P): GrpPerm, SeqEnum → GrpPerm
    • BlocksKernel(G, P): GrpPerm, SetEnum → GrpPerm
    • Example: Blocks Actions
    • Example: BlocksActions 2
  • Action on a Coset Space
    • CosetAction(G, H: parameters): Grp, Grp → Hom(Grp), GrpPerm, GrpPerm
    • RegularRepresentation(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: parameters): GrpPerm, GrpPermElt → GrpPermElt
    • JellyfishImage(G, H: parameters): GrpPerm, GrpPerm → GrpPermElt
    • JellyfishPreimage(G, x: parameters): GrpPerm, GrpPermElt → GrpPermElt
    • JellyfishPreimage(G, I: parameters): GrpPerm, GrpPerm → GrpPermElt
• Normal and Subnormal Subgroups
  • Characteristic Subgroups and Normal Series
    • DerivedSeries(G): GrpPerm → [ GrpPerm ]
    • CompositionSeries(G): GrpPerm → [ GrpPerm ]
    • CommutatorSubgroup(G): GrpPerm → GrpPerm
    • DerivedSubgroup(G): GrpPerm → GrpPerm
    • DerivedGroup(G): GrpPerm → GrpPerm
    • SolubleResidual(G): GrpPerm → GrpPerm
    • SolvableResidual(G): GrpPerm → GrpPerm
    • DerivedLength(G): GrpPerm → RngIntElt
    • LowerCentralSeries(G): GrpPerm → [ GrpPerm ]
    • NilpotencyClass(G): GrpPerm → RngIntElt
    • UpperCentralSeries(G): GrpPerm → [ GrpPerm ]
    • Centre(G): GrpPerm → GrpPerm
    • Center(G): GrpPerm → GrpPerm
    • Hypercentre(G): GrpPerm → GrpPerm
    • Hypercenter(G): GrpPerm → GrpPerm
    • pCore(G, p): GrpPerm, RngIntElt → GrpPerm
    • pCoreQuotient(G, p): GrpPerm, RngIntElt → GrpPerm, Map, GrpPerm
    • FittingGroup(G): GrpPerm → 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: Series
  • 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: Normal Subgroups
  • 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: Comp Factors
    • PrimaryAbelianInvariants(G): GrpPerm → [ RngIntElt ]
    • AbelianInvariants(G): GrpPerm → [ RngIntElt ]
    • PrimaryAbelianBasis(G): GrpPerm → [ GrpPermElt ], [ RngIntElt ]
    • AbelianBasis(G): GrpPerm → [ GrpPermElt ], [ RngIntElt ]
  • The Socle
    • Socle(G): GrpPerm → GrpPerm
    • SocleFactor(G): GrpPerm → GrpPerm
    • SocleFactors(G): GrpPerm → [ GrpPerm ]
    • SocleSeries(G): GrpPerm → [ GrpPerm ]
    • EARNS(G): GrpPerm → GrpPerm
    • AffineGeneralLinearGroup(E): GrpPerm → GrpPerm
    • AGL(E): 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: Primitive Structure
  • The Soluble Radical and its Quotient
    • Radical(G): GrpPerm → GrpPerm
    • SolubleRadical(G): GrpPerm → GrpPerm
    • SolvableRadical(G): GrpPerm → GrpPerm
    • RadicalQuotient(G): GrpPerm → GrpPerm, Hom(GrpPerm), GrpPerm
    • ElementaryAbelianSeries(G: parameters): GrpPerm → [ GrpPerm ]
    • ElementaryAbelianSeries(G, N: parameters): GrpPerm, GrpPerm → [ GrpPerm ]
    • ElementaryAbelianSeriesCanonical(G): GrpPerm → [ GrpPerm ]
    • Example: Radical
  • 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: Complements
  • Abelian Normal Subgroups
    • AbelianNormalSubgroup(G): GrpPerm → GrpPerm
    • AbelianNormalQuotient(G, H): GrpPerm, 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
    • DoubleCosetCanonical(G, H, g, K: parameters): GrpPerm, GrpPerm, GrpPermElt, 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
    • CosetTable(G, f): Grp, Map → Map
  • Transversals
    • Transversal(G, H): GrpPerm, GrpPerm → { @ GrpPermElt  @}, Map
    • RightTransversal(G, H): GrpPerm, GrpPerm → { @ GrpPermElt  @}, 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)
    • FPGroup(G, N): GrpPerm, 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
    • WordInGenerators(G, g : parameters): GrpPerm, GrpPermElt → GrpFPElt
• Automorphism Groups
  • AutomorphismGroup(G: parameters): GrpPerm → GrpAuto
  • IsIsomorphic(G, H: parameters): GrpPerm, GrpPerm → BoolElt, Hom(Grp)
  • Example: Automorphisms
• 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
  • NextExtension(P): Process → GrpFP
  • SplitExtension(G, M, F): GrpPerm, ModRng, GrpFP → GrpFP
  • Example: Cohomology
  • Example: Cohomology 2
• 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: G Module
• 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 : parameters): GrpPerm → BoolElt
    • TwoTransitiveGroupIdentification(G): GrpPerm → Tup
    • IsEven(G): GrpPerm → BoolElt
    • RecogniseAlternatingOrSymmetric(G : parameters): Grp, RngIntElt → BoolElt, Map, Map, Map, Map, BoolElt
    • AlternatingOrSymmetricElementToWord(G, g): Grp, GrpElt → BoolElt, GrpSLPElt
    • Example: Recognise Altsym2
    • 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: Recognise Altsym2
• 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
    • SolvableSchreier(G: parameters): GrpPerm
    • Verify(G: parameters ): GrpPerm
    • Example: BSGS
    • Example: BSFS 2
  • Defining Values for Attributes
    • AssertAttribute(G, "Order", n): GrpPerm, MonStgElt, RngIntElt
    • AssertAttribute(G, "Order", Q): GrpPerm, MonStgElt, [<RngIntElt, RngIntElt>]
    • AssertAttribute(G, "BSGS", S): GrpPerm, MonStgElt, GrpPermBSGS
    • Example: Random Schreier
  • 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
    • BasicStabiliser(G, i): GrpPerm, RngIntElt → GrpPerm
    • BasicStabilizerChain(G): GrpPerm → [GrpPerm]
    • BasicStabiliserChain(G): GrpPerm → [GrpPerm]
    • IsMemberBasicOrbit(G, i, a): GrpPerm, RngIntElt, Elt → BoolElt
    • NumberOfStrongGenerators(G): GrpPerm → RngIntElt
    • Nsgens(G): GrpPerm → RngIntElt
    • NumberOfStrongGenerators(G, i): GrpPerm, RngIntElt → RngIntElt
    • Nsgens(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)
  • AGL(arguments)
  • AffineGeneralLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AffineGeneralLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AffineGeneralLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AGL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AGL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AGL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AffineSpecialLinearGroup(arguments)
  • ASL(arguments)
  • AffineSpecialLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AffineSpecialLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AffineSpecialLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ASL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ASL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ASL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AffineGammaLinearGroup(arguments)
  • AGammaL(arguments)
  • AffineGammaLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AffineGammaLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AffineGammaLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AGammaL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AGammaL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AGammaL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AffineSigmaLinearGroup(arguments)
  • ASigmaL(arguments)
  • AffineSigmaLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AffineSigmaLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AffineSigmaLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ASigmaL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ASigmaL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ASigmaL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AffineSymplecticGroup(arguments)
  • ASp(arguments)
  • AffineSymplecticGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AffineSymplecticGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AffineSymplecticGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ASp(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ASp(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ASp(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AffineSigmaSymplecticGroup(arguments)
  • ASigmaSp(arguments)
  • AffineSigmaSymplecticGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • AffineSigmaSymplecticGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • AffineSigmaSymplecticGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ASigmaSp(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ASigmaSp(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ASigmaSp(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralLinearGroup(arguments)
  • PGL(arguments)
  • ProjectiveGeneralLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialLinearGroup(arguments)
  • PSL(arguments)
  • ProjectiveSpecialLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGammaLinearGroup(arguments)
  • PGammaL(arguments)
  • ProjectiveGammaLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGammaLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGammaLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGammaL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGammaL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGammaL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaLinearGroup(arguments)
  • PSigmaL(arguments)
  • ProjectiveSigmaLinearGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaLinearGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaLinearGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaL(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaL(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaL(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralUnitaryGroup(arguments)
  • PGU(arguments)
  • ProjectiveGeneralUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralUnitaryGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralUnitaryGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGU(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGU(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGU(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialUnitaryGroup(arguments)
  • PSU(arguments)
  • ProjectiveSpecialUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialUnitaryGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialUnitaryGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSU(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSU(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSU(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGammaUnitaryGroup(arguments)
  • PGammaU(arguments)
  • ProjectiveGammaUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGammaUnitaryGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGammaUnitaryGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGammaU(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGammaU(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGammaU(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaUnitaryGroup(arguments)
  • PSigmaU(arguments)
  • ProjectiveSigmaUnitaryGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaUnitaryGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaUnitaryGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaU(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaU(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaU(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSymplecticGroup(arguments)
  • PSp(arguments)
  • ProjectiveSymplecticGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSymplecticGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSymplecticGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSp(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSp(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSp(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaSymplecticGroup(arguments)
  • PSigmaSp(arguments)
  • ProjectiveSigmaSymplecticGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaSymplecticGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSigmaSymplecticGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaSp(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaSp(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSigmaSp(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroup(arguments)
  • PGO(arguments)
  • ProjectiveGeneralOrthogonalGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGO(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGO(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGO(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroupPlus(arguments)
  • PGOPlus(arguments)
  • ProjectiveGeneralOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroupPlus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroupPlus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGOPlus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGOPlus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGOPlus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroupMinus(arguments)
  • PGOMinus(arguments)
  • ProjectiveGeneralOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroupMinus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveGeneralOrthogonalGroupMinus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PGOMinus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PGOMinus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PGOMinus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroup(arguments)
  • PSO(arguments)
  • ProjectiveSpecialOrthogonalGroup(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroup(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSO(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSO(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSO(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroupPlus(arguments)
  • PSOPlus(arguments)
  • ProjectiveSpecialOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroupPlus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroupPlus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSOPlus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSOPlus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSOPlus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroupMinus(arguments)
  • PSOMinus(arguments)
  • ProjectiveSpecialOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroupMinus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSpecialOrthogonalGroupMinus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSOMinus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSOMinus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSOMinus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmega(arguments)
  • POmega(arguments)
  • ProjectiveOmega(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmega(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmega(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • POmega(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • POmega(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • POmega(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmegaPlus(arguments)
  • POmegaPlus(arguments)
  • ProjectiveOmegaPlus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmegaPlus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmegaPlus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • POmegaPlus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • POmegaPlus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • POmegaPlus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmegaMinus(arguments)
  • POmegaMinus(arguments)
  • ProjectiveOmegaMinus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmegaMinus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveOmegaMinus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • POmegaMinus(n, q): RngIntElt, RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • POmegaMinus(n, K): RngIntElt, FldFin → GrpPerm, {@ ModTupFldElt  @}
  • POmegaMinus(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSuzukiGroup(arguments)
  • PSz(arguments)
  • ProjectiveSuzukiGroup(q): RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSuzukiGroup(K): FldFin → GrpPerm, {@ ModTupFldElt  @}
  • ProjectiveSuzukiGroup(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • PSz(q): RngIntElt → GrpPerm, {@ ModTupFldElt  @}
  • PSz(K): FldFin → GrpPerm, {@ ModTupFldElt  @}
  • PSz(V): ModTupRng → GrpPerm, {@ ModTupFldElt  @}
  • AffineGroup(M): GrpMat[FldFin] → GrpPerm, {@ ModTupFldElt  @}
• Permutation Group Databases
• Ordering of Permutation Groups
  • CanonicalGenerators(G): GrpPerm → SeqEnum[GrpPermElt], GrpPermElt
  • CanonicalInvariant(G): GrpPerm → SeqEnum[SeqEnum], Seqenum
  • Example: Canonical Generators
• 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
    • NumberOfCells(P): StkPtnOrd → RngIntElt
    • CellNumber(P, h, x): StkPtnOrd, RngIntElt, RngIntElt → RngIntElt
    • CellNumber(P, x): StkPtnOrd, RngIntElt → RngIntElt
    • CellSize(P, h, i): StkPtnOrd, RngIntElt, RngIntElt → RngIntElt
    • CellSize(P, i): StkPtnOrd, RngIntElt → RngIntElt
    • Cell(P, h, i): StkPtnOrd, RngIntElt, RngIntElt → SeqEnum
    • Cell(P, i): StkPtnOrd, RngIntElt → SeqEnum
    • Random(P, i): StkPtnOrd, RngIntElt → RngIntElt
    • Representative(P, i): StkPtnOrd, RngIntElt → RngIntElt
    • Rep(P, i): StkPtnOrd, RngIntElt → RngIntElt
    • ParentCell(P, i): StkPtnOrd, RngIntElt → RngIntElt
  • Operations on Ordered Partition Stacks
    • SplitCell(P, i, x): StkPtnOrd, RngIntElt, RngIntElt → BoolElt
    • SplitCell(P, i, Q): StkPtnOrd, RngIntElt, SeqEnum[RngIntElt] → BoolElt
    • SplitAllByValues(P, V): StkPtnOrd, SeqEnum[RngIntElt] → BoolElt, RngIntElt
    • SplitCellsByValues(P, C, V): StkPtnOrd, SeqEnum[RngIntElt], SeqEnum[RngIntElt] → BoolElt, RngIntElt
    • SplitCellsByValues(P, i, V): StkPtnOrd, RngIntElt, SeqEnum[RngIntElt] → BoolElt, RngIntElt
    • Pop(P): StkPtnOrd
    • Pop(P, h): StkPtnOrd, RngIntElt
    • Advance(X, L, P, h): StkPtnOrd, seqEnum[RngIntElt], StkPtnOrd, RngIntElt
    • Example: Ordered Partition Stack