Periodic normal subgroups of linear groups

In a yet to be published work R. E. Phillips and J. G. Rainbolt prove that every image of a periodic linear group (of finite degree) with trivial unipotent radical is isomorphic to a linear group over the same field and of bounded degree. Here we offer an alternative proof that is both quits short and delivers a little more. Our basic theorem, from which follow a number of corollaries, is the following. There is an integer-valued function f(n) of n only such that if G is any linear group of finite degree n and characteristic p greater than or equal to 0 and if N is any periodic normal subgroup of G, with O-p(N)= (1) if p not equal 0, then GIN is isomorphic to a linear group of degree f(n) and characteristic p. One corollary is Phillips and Rainbolt's Theorem. A second has the condition O-p(N) = (1) if p +/-0 replaced by O-p(G)less than or equal to N if p +/-0, but with the same conclusion land with the same function f(n)).