On a conjecture on simple groups

Abstract

The purpose of this paper is to rephrase a conjecture about simple groups into the language of linear algebra. Let G be a group of finite order o(G). Then by rF we shall mean the group ring of G over a field of characteristic p (for instance the integers modulo p). We shall denote the radical of rF by N,. If p = 0 or p o(G), then it is known that Np=(O); and if p|o(G), Np (O). We now consider the following two assertions: (A) If G is a simple group of odd order, o(G) is a prime. (B) If G is a group of odd order o(G), then for some prime p, p[ o(G), we can find a gCG, g1, such that g-1CNp. The theorem which we propose to prove is: