Involutory Automorphisms of Groups of odd Order and Their Fixed Point Groups

Abstract

Let G be a finite group of odd order with an automorphism θ of order 2. (We use without further reference the fact, established by W. Feit and J. G. Thompson, that all groups of odd order are soluble.) Let Gθ denote the subgroup of G formed by the elements fixed under θ. It is an elementary result that if Gθ = 1 then G is abelian. But if we merely postulate that Gθ be cyclic, the structure of G may be considerably more complicated—indeed G may have arbitrarily large soluble length.