The Fitting length of finite soluble groups II Fixed-point-free automorphisms

Let G be a finite soluble group, and let h(G) be the Fitting length of G. If yo is a fixed-point-free automorphism of G, that is C-G(phi) {1}, we denote by W (phi) the composition length of <phi >. A long-standing conjecture is that h(G) <= W (phi), and it is known that this bound is always true if the order of G is coprime to the order of phi. In this paper we find some bounds to h(G) in function of W(phi) without assuming that (vertical bar G vertical bar, vertical bar phi vertical bar) = 1. In particular we prove the validity of the "universal" bound h(G) < 7W(phi)(2). This improves the exponential bound known earlier from a special case of a theorem of Dade. (C) 2017 Elsevier Inc. All rights reserved.