J Austral Math Soc Ser A 53 pp116--119, 1992.

Periodic Groups with Many Permutable Subgroups

Patrizia Longobardi, Mercede Maj, Akbar Rhemtulla and Howard Smith

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 gert the following characterization. A finitely generated group G is centre-by-finite if and only if every infinite aet of subgroups of G contains a pair that permute.

1991 AMS Subject Classification: 20E15

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Authors

Patriza Longobardi

Dipartimento di Matematica e, Applicazioni, Università degli Studi de Napoli, Via Mezzocannone 8, 80124, Naples, Italy.

Mercede Maj

Dipartimento di Matematica e, Applicazioni, Università degli Studi de Napoli, Via Mezzocannone 8, 80124, Naples, Italy.

Akbar Rhemtulla

Department of Mathematics, University of Albertam Edmonton, Canada, T6G 2G1.

Howard Smith

Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837, USA.