Automorphism Groups

• Introduction
• Creation of Automorphism Groups
  • AutomorphismGroup(G): Grp → GrpAuto
  • Example: Automorphisms of a Non-Soluble Matrix Group
  • AutomorphismGroup(G, Q, I): Grp, SeqEnum[GrpElt], SeqEnum[SeqEnum[GrpElt]] → GrpAuto
• Access Functions
  • Group(A): GrpAuto → Grp
  • NumberOfGenerators(A): GrpAuto → RngIntElt
  • Ngens(A): GrpAuto → RngIntElt
  • NumberOfPCGenerators(A): GrpAuto → RngIntElt
  • NPCGenerators(A): GrpAuto → RngIntElt
  • NPCgens(A): GrpAuto → RngIntElt
  • Generators(A): GrpAuto → SetEnum
  • PCGenerators(A): GrpAuto → SetIndx
  • InnerGenerators(A): GrpAuto → SeqEnum
  • CharacteristicSeries(A): GrpAuto → SeqEnum
  • IsSoluble(A): GrpAuto → BoolElt
  • IsSolvable(A): GrpAuto → BoolElt
  • IsSolubleAutomorphismGroupPGroup(A): GrpAuto → BoolElt
  • IsSolvableAutomorphismGroupPGroup(A): GrpAuto → BoolElt
• Order Functions
  • Order(A): GrpAuto → RngIntElt
  • # A: GrpAuto → RngIntElt
  • FactoredOrder(A): GrpAuto → [ <RngIntElt, RngIntElt> ]
  • OuterOrder(A): GrpAuto → RngIntElt
  • Example: Automorphism Group of PGL(2, 9)
• Representations of an Automorphism Group
  • PermutationRepresentation(A): GrpAuto → Map, GrpPerm, SetIndx
  • PermutationGroup(A): GrpAuto → GrpPerm
  • PermutationSupport(A): GrpAuto → SetIndx
  • PCGroupAutomorphismGroupPGroup(A): GrpAuto → BoolElt, Map, GrpPC
  • FPGroup(A): GrpAuto → GrpFP, Map
  • OuterFPGroup(A): GrpAuto → GrpFP, Map
  • Example: Permutation Representation and Presentation
  • Example: Automorphisms of the Higman-Sims Group
• Automorphisms
  • A . i: GrpAuto, RngIntElt → GrpAutoElt
  • Identity(A): GrpAuto → GrpAutoElt
  • Id(A): GrpAuto → GrpAutoElt
  • A ! 1: GrpAuto, RngIntElt → GrpAutoElt
  • A ! f: GrpAuto, Map → GrpAutoElt
  • Order(f): GrpAutoElt → RngIntElt
  • f * g: GrpAutoElt, GrpAutoElt → GrpAutoElt
  • f ^ n: GrpAutoElt, RngIntElt → GrpAutoElt
  • (g₁, ..., gᵣ): GrpAutoElt, ..., GrpAutoElt → GrpAutoElt
  • g eq h: GrpAutoElt, GrpAutoElt → BoolElt
  • g ne h: GrpAutoElt, GrpAutoElt → BoolElt
  • IsInner(f): GrpAutoElt → BoolElt, GrpElt
  • Example: Arithmetic with Automorphisms
  • Example: Characteristic Subgroups
• Stored Attributes of an Automorphism Group
  • HasAttribute(A, s): GrpAuto, MonStgElt → BoolElt, .
  • AssertAttribute(A, s, v): GrpAuto, MonStgElt, .
  • Example: Automorphisms of a Group of Order 904
  • Example: Outer Automorphism Group of A6
• Holomorphs
  • Holomorph(G): Grp → GrpPerm, HomGrp, HomGrp
  • Holomorph(GrpFP, G): Grp → Cat[GrpFP], GrpFP, HomGrp, HomGrp
  • Holomorph(G, A): Grp, GrpAuto → GrpPerm, HomGrp, HomGrp
  • Holomorph(GrpFP, G, A): Cat[GrpFP], Grp, GrpAuto → GrpFP, HomGrp, HomGrp
  • Example: Holomorph of PGL(2, 9)
• Bibliography