Groups Admitting an Automorphism of Prime Order with Finite Centralizer

Abstract

We consider a group G with an automorphism of finite, usually prime, order. If G has finite Hirsch number, and also if G satisfies various stronger rank restrictions, we study the consequences and equivalent hypotheses of having only finitely many fixed-points. In particular we prove that if a group G with finite Hirsch number $${\mathfrak{h}}$$ admits an automorphism $${\varphi}$$ of prime order p such that $${\vert C_{G}(\varphi) \vert = n < \infty,}$$ then G has a subgroup of finite index bounded in terms of p, n and $${\mathfrak{h}}$$ that is nilpotent of p-bounded class.