Abstract

In 1959, Thompson proved that a group with a fixed point free automorphism of prime order must be nilpotent. G. Higman had already asked whether one could say anything about their nilpotency class. We survey the literature on this question and give an improvement on the bound for the nilpotency class.

Introduction

Thompson proved, in his thesis of 1959, that groups with fixed point free automorphisms of prime order are nilpotent. Such groups are exactly the Frobenius kernels and Frobenius had conjectured that they must be nilpotent. This conjecture was informed by two fairly simple results; a group with a fixed point free automorphism of order 2 is abelian and a group with a fixed point free automorphism of order 3 is nilpotent. Proofs may be found in Chapter 10 of.

This last result can, via a simple argument, be strengthened to give that a group with a fixed point free automorphism of order 3 is nilpotent of class at most 2. This was first published by Neumann, see and. This led G. Higman to ask in of 1959 whether a group with a fixed point free automorphism of order p must be of nilpotency class at most some number, now called h(p) in honor of Higman. He proved that in fact such a number does exist.