Finite group actions on abelian groups of finite Morley rank

TL;DR

This paper investigates finite group actions on abelian groups within the finite Morley rank framework, providing results that support broader conjectures and removing previous assumptions in the field.

Contribution

It introduces new general results on finite group actions in the finite Morley rank setting, crucial for removing the 'sharpness' assumption and proving linearity conjectures.

Findings

Proves the linearity of irreducible definable actions of simple algebraic groups on elementary abelian p-groups.

Develops key results linking finite group actions to problems in finite Morley rank groups.

Supports the removal of the 'sharpness' assumption in existing Morley rank group theories.

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 Ay\c{s}e 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].