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 tr ivial unipotent radical is isomorphic to a linear group over the same field and of bounded degree. Here we offer an alternative proof that i s both quits short and delivers a little more. Our basic theorem, from which follow a number of corollaries, is the following. There is an i nteger-valued function f(n) of n only such that if G is any linear gro up 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 n ot equal 0, then GIN is isomorphic to a linear group of degree f(n) an d 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)le ss than or equal to N if p +/-0, but with the same conclusion land wit h the same function f(n)).