Solvable groups admitting a fixed-point-free automorphism of prime power order

Abstract

The purpose of the paper is to prove the following: Theorem 1. Suppose G is a finite group which admits an automorphism a of order pn, where p is an odd prime, such that a fixes only the identity element of G.(aFurthermore, both these inequalities are best-possible.Here h(G), the Fitting height (also called the nilpotent length) of G, is as defined in [7].P(G), the 7t-length of G, is defined in an obvious analogy to the definition of ^-length in [2].Higman [3] proved Theorem 1 in the case w = l (subsequently, without making any assumptions on the solvability of G, Thompson [6] obtained the same result).Hoffman [4] and Shult [5] proved Theorem 1 provided that either p is not a Fermat prime or a Sylow 2-group of G is abelian.For p = 2, Gorenstein and Herstein[l] obtained Theorem 1 if »S2, and Hoffman and Shult both obtained Theorem 1 provided that a Sylow g-group of G is abelian for all Mersenne primes q which divide the order of G. Shult, who considers a more general situation of which Theorem 1 is a special case, recently extended his results to include all primes, but his bound on h(G) is not best-possible in the special case of Theorem 1.It also should be noted that Thompson [7] obtained a bound for h(G) under a much more general hypothesis than that considered in the other papers mentioned.Theorem 1 is a consequence of Theorem 2. Let G be a finite group admitting a fixed-point-free automorphism a of order pn, p an odd prime, and let H be a normal Hall subgroup of G such that H contains its centralizer in G. Then the automorphism of G/H induced by