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 $p$-groups of finite Morley rank [16, Conjecture 12].

Expanded and corrected in comparison with previous versions

References in corpus (3)

• Groups of finite Morley rank with a generically multiply transitive action on an abelian group
• Linear groups of finite Morley rank
• Automorphism groups of $P^1$-bundles over a non-uniruled base

Cited by in corpus (1)

• Groups of finite Morley rank with a generically multiply transitive action on an abelian group