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

Let G be a finite soluble group, and let be the Fitting length of G. If φ is a fixed-point-free automorphism of G, that is , we denote by the composition length of . A long-standing conjecture is that , 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 in function of without assuming that . In particular we prove the validity of the “universal” bound . This improves the exponential bound known earlier from a special case of a theorem of Dade.