On Almost Fixed Point Free Automorphisms

Ideas from the theory of p-adic analytic groups are employed in the study of finite groups having automorphisms with few fixed points. In particular we prove that the derived length of a finite p-group admitting an automorphism of order p^k with p^m fixed points is bounded in terms of p, m, k. This extends a well-known result of Alperin dealing with the case k = 1. Some applications to soluble groups are also given.