Periodic normal subgroups of linear groups

Abstract

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 quite 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≥ 0 and if N is any periodic normal subgroup of G, with \(O_p(N) = \langle 1 \rangle\) if \(p \neq 0\), then G/N 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 \(O_p(N) = \langle 1 \rangle\) if \(p \neq 0\) replaced by \(O_p(G) \leq N\) if \(p \neq 0\), but with the same conclusion (and with the same function f (n).