Exponent of a finite group of odd order with an involutory automorphism

摘要

Let G be a finite group of odd order admitting an involutory automorphism f. We obtain two results bounding the exponent of [G, f]. Denote by G-f the set {[g, f] | g. G} and by Gf the centralizer of f, that is, the subgroup of fixed points of f. The obtained results are as follows. 1. Assume that the subgroup x, y has derived length at most d and x e = 1 for every x, y. G-f. Suppose that Gf is nilpotent of class c. Then the exponent of [G, f] is (c, d, e)-bounded. 2. Assume that Gf has rank r and x e = 1 for each x. G-f. Then the exponent of [G, f] is (e, r)-bounded.