The Fitting length of finite soluble groups II

Let G be a finite soluble group, and let h(G) be the Fitting length of G. If φ is a fixed-point-free automorphism of G, that is CG(φ)={1}, we denote by W(φ) the composition length of 〈φ〉. A long-standing conjecture is that h(G)≤W(φ), and it is known that this bound is always true if the order of G is coprime to the order of φ. In this paper we find some bounds to h(G) in function of W(φ) without assuming that (|G|,|φ|)=1. In particular we prove the validity of the “universal” bound h(G)<7W(φ)2. This improves the exponential bound known earlier from a special case of a theorem of Dade.