Black-Box Groups

• Introduction
• Construction of an SLP-Group and its Elements
  • Structure Constructors
    • NaturalBlackBoxGroup(H): Grp → GrpBB
  • Construction of an Element
    • Identity(G): GrpBB → GrpBBElt
    • Id(G): GrpBB → GrpBBElt
    • G ! 1: GrpBB, RngIntElt → GrpBBElt
• Arithmetic with Elements
  • u * v: GrpBBElt, GrpBBElt → GrpBBElt
  • u ^ m: GrpBBElt, RngIntElt → GrpBBElt
  • u ^ v: GrpBBElt, GrpBBElt → GrpBBElt
  • (u, v): GrpBBElt, GrpBBElt → GrpBBElt
  • Accessing the Defining Generators
    • G . i: GrpBB, RngIntElt → GrpBBElt
    • Generators(G): GrpBB → { GrpBBElt}
    • NumberOfGenerators(G): GrpBB → RngIntElt
    • Ngens(G): GrpBB → RngIntElt
• Operations on Elements
  • Equality and Comparison
    • u eq v: GrpBBElt, GrpBBElt → BoolElt
    • u ne v: GrpBBElt, GrpBBElt → BoolElt
  • Attributes of Elements
    • Parent(u): GrpBBElt → GrpBB
    • UnderlyingElement(u): GrpBBElt → GrpElt
    • Order(u): GrpBBElt → RngIntElt
    • Example: Standard Generators of M24
• Set-Theoretic Operations
  • Membership and Equality
    • g in G: GrpBBElt, GrpBB → BoolElt
  • Set Operations
    • PseudoRandom(G): GrpBB → GrpBBElt
    • Rep(G): GrpBB → GrpBBElt
  • Coercions Between Related Groups
    • G ! g: GrpBB, GrpBBElt → GrpBBElt