Groups Admitting an Automorphism of Prime Order with Finite Centralizer

We consider a group G with an automorphism of finite, usu- ally 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 h admits an automorphism ϕ of prime order p such that |CG (ϕ)| = n < ∞, then G has a subgroup of finite index bounded in terms of p, n and h that is nilpotent of p-bounded class.