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Finite group actions on abelian groups of finite Morley rank

Alexandre Borovik

Abstract

This paper develops some general results about actions of finite groups on (infinite) abelian groups in the finite Morley rank category. They are linked to a range of problems on groups of finite Morley rank discussed in [16]. Crucially, these results are needed for the forthcoming work by Ayşe Berkman and myself [5] where we remove the `sharpness' assumption from [4]. Also, they yield a proof of the long standing conjecture of linearity of irreducible definable actions of simple algebraic groups on elementary abelian $p$-groups of finite Morley rank [16, Conjecture 12].

Finite group actions on abelian groups of finite Morley rank

Abstract

This paper develops some general results about actions of finite groups on (infinite) abelian groups in the finite Morley rank category. They are linked to a range of problems on groups of finite Morley rank discussed in [16]. Crucially, these results are needed for the forthcoming work by Ayşe Berkman and myself [5] where we remove the `sharpness' assumption from [4]. Also, they yield a proof of the long standing conjecture of linearity of irreducible definable actions of simple algebraic groups on elementary abelian -groups of finite Morley rank [16, Conjecture 12].

Paper Structure

This paper contains 36 sections, 14 theorems, 42 equations.

Table of Contents

  1. A more technical introduction
  2. Ranked universes
  3. Some terminology for group actions
  4. Finite groups and Jordan properties
  5. From finite groups to simple algebraic groups over algebraically closed fields
  6. Linearisation of actions of simple algebraic groups
  7. Linear groups of finite Morley rank
  8. Linearisation of actions of solvable-by-finite groups
  9. Enveloping algebras enter the scene
  10. Definitions and generalities
  11. The case of smooth action
  12. The weight decomposition for a coprime action of a finite abelian group and the Multiplicity Formula
  13. Proof of Theorem \ref{['th:lower bound']}
  14. Theorem \ref{['th:lower bound']}: comments
  15. Proof of Theorem \ref{['th:-corollary-via-LP']} and Jordan properties
  16. ...and 21 more sections

Key Result

Theorem 1

Under the assumptions of Problem prob:lower bound,

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1.1
  • Theorem 5
  • Lemma 2.1
  • proof
  • Lemma 2.3
  • proof
  • ...and 15 more