Abstract

Let A be a finite group acting coprimely on a finite group G. It was recently discovered that the exponent of CG(A) may have strong impact over the exponent of G. In this paper we discuss results on the exponent of a group with coprime automorphisms, as well as some applications and open problems. No detailed proofs are given.

Introduction

Let A be a finite group acting coprimely on a finite group G. It is well-known that the structure of the centralizer CG(A) (the fixed-point subgroup) of A has strong influence over the structure of G. To exemplify this we mention the following results.

The celebrated theorem of Thompson [27] says that if A is of prime order and CG(A) = 1, then G is nilpotent. On the other hand, any nilpotent group admitting a fixed-point-free automorphism of prime order q has nilpotency class bounded by some function h(q) depending on q alone. This result is due to Higman [13]. The reader can find in [15] and [16] an account on the modern developments related to Higman's theorem. The next result is a consequence of the classification of finite simple groups [29]: If A is a group of automorphisms of G whose order is coprime to that of G and CG(A) is nilpotent or has odd order, then G is soluble.