×

Periodic groups with many permutable subgroups. (English) Zbl 0768.20015

If each infinite set of elements of \(G\) contains a pair which commute then \(G\) is centre-by-finite [B. H. Neumann, ibid. 21, 467-472 (1976; Zbl 0333.05110)]. Groups in which each infinite set of subgroups of \(G\) contains a pair which permute were considered by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold [ibid. 48, 397-401 (1990; Zbl 0705.20030)] who called them \(PH\)-groups. They showed that a finitely generated soluble \(PH\)-group is centre-by-finite and that torsion-free \(PH\)-groups are abelian. Here it is shown that a periodic \(PH\)-group is locally finite and that a finitely generated group is a \(PH\)-group if and only if it is centre-by-finite. The earlier paper contains an example of a \(PH\)-group which is not centre-by-finite.

MSC:

20F50 Periodic groups; locally finite groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F24 FC-groups and their generalizations
PDFBibTeX XMLCite