Locally soluble infinite groups in which every element has prime power order.
A periodic group \(G\) is a CP-group if every element has prime power order or, equivalently, the centralizer of every element is a \(p\)-group for some prime \(p\). Finite CP-groups were studied in a paper of \textit{G. Higman} [J. Lond. Math. Soc. 32, 335-342 (1957; Zbl 0079.03204)] and this work was continued by \textit{W. Shi} and \textit{W. Yang} [J. Yunnan Ed. Coll. 1, 2-10 (1986)]. In the paper under review the authors study infinite CP-groups. Of course there are rather exotic infinite CP-groups so for the most part the authors study locally soluble such groups. In this case they prove that if \(G\) is a CP-group then it is a \(\{p,q\}\)-group, for primes \(p,q\), and they obtain structure theorems for such locally soluble groups with min-n (respectively max-n).
scientific article; zbMATH DE number 2067242