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Periodic groups with many permutable subgroups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Patrizia Longobardi ,
Mercede Maj ,
Akbar Rhemtulla  and
Howard Smith
Patrizia Longobardi
Affiliation:
Dipartimento di Matematica e ApplicazioniUniversità degli Studi de NapoliVia Mezzocannone 8, 80134 Naples, Italy
Mercede Maj
Affiliation:
Dipartimento di Matematica e ApplicazioniUniversità degli Studi de NapoliVia Mezzocannone 8, 80134 Naples, Italy
Akbar Rhemtulla
Affiliation:
Department of MathematicsUniversity of AlbertaEdmontonCanada, T6G 2G1
Howard Smith
Affiliation:
Department of MathematicsBucknell UniversityLewisburg Pennsylvania 17837, USA
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Abstract

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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.

MSC classification

Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 1 , August 1992 , pp. 116 - 119
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Curzio, Mario, Lennox, John, Rhemtulla, Akbar and Wiegold, James, ‘Groups with many permutable subgroups’, J. Austral Math. Soc. (to appear).Google Scholar
[2]Robinson, D. J. S., Finiteness conditions and generalized soluble groups, 1 (Springer Verlag, 1972).Google Scholar
[3]Tomkinson, M. J., FC-groups, (Research Notes in Math No. 96, Pitman, 1984).Google Scholar