On the rank of a finite group of odd order with an involutory automorphism

Abstract

Let G be a finite group of odd order admitting an involutory automorphism $$\phi $$ , and let $$G_{-\phi }$$ be the set of elements of G transformed by $$\phi $$ into their inverses. Note that $$[G,\phi ]$$ is precisely the subgroup generated by $$G_{-\phi }$$ . Suppose that each subgroup generated by a subset of $$G_{-\phi }$$ can be generated by at most r elements. We show that the rank of $$[G,\phi ]$$ is r-bounded.