  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].
