Periodic groups with many permutable subgroups
Patrizia Longobardi, Mercede Maj, A. H. Rhemtulla, Howard Smith · Journal of the Australian Mathematical Society Series A Pure Mathematics and Statistics · 1992
Abstract Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.