Matrix Groups over Infinite Fields

• Overview
• Construction of Congruence Homomorphisms
  • CongruenceImage(G : parameters): GrpMat → GrpMat, HomGrp, []
• Testing Finiteness
  • IsFinite(G : parameters): GrpMat → BoolElt, RngIntElt
  • IsomorphicCopy(G : parameters): GrpMat → BoolElt, GrpMat, HomGrp
  • Order(G : parameters): GrpMat → RngIntElt
• Deciding Virtual Properties of Linear Groups
  • IsSolubleByFinite(G : parameters): GrpMat → BoolElt
  • IsPolycyclicByFinite(G : parameters): GrpMat → BoolElt
  • IsNilpotentByFinite(G : parameters): GrpMat → BoolElt
  • IsAbelianByFinite(G : parameters): GrpMat → BoolElt
  • IsCentralByFinite(G : parameters): GrpMat → BoolElt
• Hirsch Number and Prüfer Rank
  • HasFiniteRank(G): GrpMat → BoolElt
  • HirschNumber(G): GrpMat → RngIntElt
  • HasFiniteIndex(G, H): GrpMat, GrpMat → BoolElt
  • PrueferRankBound(G): GrpMat → BoolElt
• Other Properties of Linear Groups
  • IsCompletelyReducible(G : parameters): GrpMat → BoolElt
  • CompletelyReduciblePart(G): GrpMat → GrpMat, GrpMatElt
  • IsUnipotent(G): GrpMat → BoolElt, GrpMatElt
  • IsNilpotent(G): GrpMat → BoolElt
  • IsSoluble(G : parameters): GrpMat → BoolElt
  • IsPolycyclic(G : parameters): GrpMat → BoolElt
  • HasFiniteOrder(g : parameters ): GrpMatElt → BoolElt, RngIntElt
• Other Functions for Nilpotent Matrix Groups
  • RecogniseAbelian(G): GrpMat → GrpGPC, Map, Map
  • SylowSystem(G : parameters): GrpMat[FldFin] → []
  • IsIrreducibleFiniteNilpotent(G : parameters): GrpMat → BoolElt, Any
  • IsPrimitiveFiniteNilpotent(G : parameters): GrpMat → BoolElt, Any
• Examples
  • Example: Finite Group over a Rational Function Field
  • Example: Finite Group in Characteristic 5
  • Example: Infinite Group in Characteristic 5
  • Example: Infinite Integral Matrix Group
  • Example: Infinite Group over a Quadratic Field
  • Example: Infinite Group over a Quartic Number Field
  • Example: Group over an Algebraic Function Field
  • Example: Finite Group in Characteristic 2
  • Example: Hirsch Number and Finite Index
  • Example: Completely Reducible Part of a Group
  • Example: Imprimitive Finite Nilpotent Group
  • Example: Recognising an Abelian Matrix Group
  • Example: Sylow System of a Nilpotent Group
  • Example: Infinite Nilpotent Group over a Function Field
• Bibliography