Almost fixed-point-free automorphisms of prime order.
The author discusses groups \(G\) possessing an automorphism \(\phi\) of prime order \(p\) in which \(C_G(\phi)\) is finite of order \(n\). In this context the author discusses groups of finite Hirsch number. A group \(G\) has finite Hirsch number if it has a series of finite length such that each factor of the series is either locally finite or infinite cyclic, the Hirsch number being the number of infinite cyclic factors in this series. As an example of some of the results, the author proves that such a group \(G\) with such an automorphism has a soluble characteristic subgroup of finite index of \((p,n)\)-bounded derived length. He also proves that if \(G\) is soluble of finite rank with such an automorphism then \(G\) has a characteristic subgroup of finite index that is nilpotent of class at most a function of \(p\).
characteristic subgroups of finite index
scientific article; zbMATH DE number 5997907