Longobardi, Patrizia; Maj, Mercede; Rhemtulla, Akbar; Smith, Howard Periodic groups with many permutable subgroups. (English) Zbl 0768.20015 J. Aust. Math. Soc., Ser. A 53, No. 1, 116-119 (1992). 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. Reviewer: M.J.Tomkinson (Glasgow) Cited in 1 Document MSC: 20F50 Periodic groups; locally finite groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20F24 FC-groups and their generalizations Keywords:permutable subgroups; infinite set of subgroups; finitely generated soluble \(PH\)-groups; periodic \(PH\)-group; locally finite groups; finitely generated groups; centre-by-finite groups Citations:Zbl 0333.05110; Zbl 0705.20030 PDFBibTeX XMLCite \textit{P. Longobardi} et al., J. Aust. Math. Soc., Ser. A 53, No. 1, 116--119 (1992; Zbl 0768.20015)