Almost regular automorphisms of finite groups of bounded rank (Q1357984)

It is proved that if a finite or solvable group of rank \(r\) admits an automorphism of order coprime to the order of the group with exactly \(m\) fixed points, then the group has a solvable subgroup of \((r,m)\)-bounded index whose derived length is bounded in terms of \(r\) only. Earlier \textit{A. Shalev} proved under the same assumptions that the group has a solvable subgroup of \((r,m)\)-bounded index whose derived length is \((r,m)\)-bounded [Isr. J. Math. 82, No. 1-3 395-404 (1993; Zbl 0817.20026)]. The proof is reduced to the case of finite \(p\)-groups. It is based on using Shalev's above-mentioned result and the theory of powerful \(p\)-groups.