Let \(G\) be a periodic group. An element \(x\) of \(G\) is said to be ‘\(M\)-permutable’ if the cyclic subgroup \(\langle x\rangle\) is permutable with all maximal \(p\)-subgroups of \(G\) for all prime numbers \(p\). The ‘generalized centre’ of \(G\) is the subgroup \(gz(G)\) generated by all \(M\)-permutable elements of \(G\). The ‘generalized upper central series’ and the ‘generalized hypercentre’ \(gz_\infty(G)\) can then be defined in analogy to the ordinary upper central series and hypercentre.
In the paper under review, the authors study properties of the generalized hypercentre of periodic groups. In particular, the main result states that if \(G\) is a hyperfinite group (i.e., if \(G\) has an ascending normal series with finite factors) and \(H\) is a hypercyclic subgroup of \(G\), then the product \(Hgz_\infty(G)\) is hypercyclic. In particular the generalized hypercentre of any hyperfinite group is hypercyclic.