Extensions of periodic linear groups with finite unipotent radical

A group is called p-linear if it is isomorphic to a subgroup of GL(n,K) for some field K of characteristic p and some integer n. Let H be a normal subgroup of G and assume that both H and G/H are periodic and p-linear. In addition, assume that both H and G/H have finite unipotent radicals and that the Hirsch-Plotkin radical of H is Cernikov. The main result of this article is a proof that under these assumptions G is p-linear. An example is provided showing the result is false if the assumption regarding the Hirsch-Plotkin radical is removed.