Involutory automorphisms of groups of odd order

Abstract

Let G be a finite group of odd order with an automorphism ω of order 2. The Feit-Thompson theorem implies that G is soluble and this is assumed throughout the paper. Let Gω denote the subgroup of G consisting of those elements fixed by ω. If F(G) denotes the Fitting subgroup of G then the upper Fitting series of G is defined by F1(G) = F(G) and Fr+1(G) = the inverse image in G of F(G/Fr(G)). G(r) denotes the rth derived group of G. The principal result of this paper may now be stated as follows: THEOREM 1. Let G be a group of odd order with an automorphism ω of order 2. Suppose that Gω is nilpotent, and that G(r)ω = 1. Then G(r) is nilpotent and G = F3(G).