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

Abstract

Here h(G), the Fitting height (also called the nilpotent length) of G, is as defined in [7]. 1I(G), the r-length of G, is defined in an obvious analogy to the definition of p-length in [2]. Higman [3] proved Theorem 1 in the case n =1 (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 [1] obtained Theorem 1 if n< 2, and Hoffman and Shult both obtained Theorem 1 provided that a Sylow q-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