Abstract

This paper develops some general results about actions of finite groups on infinite abelian groups of exponent $p$ in the finite Morley rank category. These results are applicable to a range of problems on groups of finite Morley rank. 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. Crucially, these results are needed for the papers by Ayşe Berkman and myself where we have proved an explicit, and best possible, upper bound for the degree of generic multiple transitivity for an action of a group of finite Morley rank on an abelian group.

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