On Almost Regular Automorphisms of Finite p-Groups

Abstract

In this paper we prove that there are functions f(p, m, n) and h(m) such that any finite p-group with an automorphism of order pn, whose centralizer has pm points, has a subgroup of derived length ⩽h(m) and index ⩽f(p, m, n). This result gives a positive answer to a problem raised by E. I. Khukhro (see also Problem 14.96 from the “Kourovka Notebook” (1999, E. I. Khukhro and V. D. Mazurov (Eds.), “The Kourovka Notebook: Unsolved Problems in Group Theory,” 14th ed., Novosibirsk)).