Almost regular automorphisms
This survey gives a detailed description of the theory of finite nilpotent groups admitting an automorphism with exactly \(m\) fixed points. The paper is divided into three sections. In section 1 the author discusses applications of theorems on Lie rings with regular automorphisms to the theory of nilpotent groups with regular automorphisms. Besides of the theorems of G. Higman, V. A. Kreknin and A. I. Kostrikin this section contains an account of Lie-ring methods such as the associated Lie ring and the Mal'cev correspondence. Section 2 is devoted to the coprime case where the order of the automorphism is coprime to the order of the group. The theorem of E. I. Khukhro on nilpotent groups with automorphism of prime order is cited as well as some of its applications and extensions. Section 3 deals with the situation where a nilpotent \(p\)-group admits an automorphism of order \(p^n\) with exactly \(p^m\) fixed points. The Alperin-Khukhro theorem, Khukhro-Shalev theorem, Medvedev's theorem and connections with \(p\)-groups of given coclass are discussed. Many results are provided with proofs or the basic ideas of the proofs. The reader can find open problems in this field.
scientific article; zbMATH DE number 1315225
scientific article