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Almost fixed-point-free automorphisms of prime order. (English) Zbl 1245.20037
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 \(p\).

20F16 Solvable groups, supersolvable groups
20E36 Automorphisms of infinite groups
20F19 Generalizations of solvable and nilpotent groups
20F50 Periodic groups; locally finite groups
20F14 Derived series, central series, and generalizations for groups
20E07 Subgroup theorems; subgroup growth
Full Text: DOI
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