This is a compilation of the lectures given in 1990–97 in the universities of Novosibirsk, Freiburg (in Breisgau), Trento and Cardiff. The book gives a concise account of several, mostly very recent, theorems on the structure of finite p-groups admitting p-automorphisms with few fixed points. The proofs, given in full detail, require various powerful methods of studying nilpotent p-groups; these methods are presented in the manner of a textbook, accessible for students with only a basic knowledge of linear algebra and group theory. Every chapter ends with exercises which vary from elementary checks to relevant results from research papers (but none of them is referred to in the proofs).

By the classical theorems of G. Higman, V. A. Kreknin and A. I. Kostrikin, a Lie ring is soluble (nilpotent) if it has a fixed-point-free automorphism of finite (prime) order. (These Lie ring theorems are also included along with all necessary preliminary material.) Prompted by and based on these Lie ring results, the main theorems of the book state that a finite p-group is close to being soluble (nilpotent) in terms of the order of a p-automorphism and the number of its fixed points. These results can be viewed as general structure theorems about finite p-groups. They are closely related to the theory of (pro-) p-groups of maximal class and given coclass and have natural extensions to locally finite p-groups.

Presenting linear (mostly Lie ring) methods in the theory of nilpotent groups is another main objective of the book.