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Groups with many permutable subgroups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Mario Curzio ,
John Lennox ,
Akbar Rhemtulla  and
James Wiegold
Mario Curzio
Affiliation:
Dipartimento di Matematica Pura e Applicata Università Degli Studi di NapoliNapoli, Italia
John Lennox
Affiliation:
School of Mathematics University of WalesCollege of Cardiff Cardiff CF2 4AG Wales
Akbar Rhemtulla
Affiliation:
Department of Mathematics University of AlbertaEdmonton, Alberta, Canada
James Wiegold
Affiliation:
School of Mathematics University of WalesCollege of Cardiff Cardiff CF2 4AG Wales
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Abstract

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We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

MSC classification

Secondary: 20F16: Solvable groups, supersolvable groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 397 - 401
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Adian, S., The Burnside problem and identities in groups, (Ergeb. Math. Grenzgeb., Springer-Verlag, Berlin, Heidelberg and New York, 1979).CrossRefGoogle Scholar
[2]Groves, J. R. J., ‘A conjecture of Lennox and Wiegold concerning supersoluble groups’, J. Austral. Math. Soc. 35 (1983), 218–220.CrossRefGoogle Scholar
[3]Heineken, Hermann and Lennox, John C., ‘A note on products of abelian groups’, Arch. Math. 41 (1983), 498–501.CrossRefGoogle Scholar
[4]Îto, N., ‘Über das Produkt von zwei abelschen Gruppen’, Math. Z. 62 (1955), 400–401.CrossRefGoogle Scholar
[5]Iwasawa, K., ‘Über die endlichen Gruppen und die Verbände ihrer Untergruppen’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 4 (1941), 171–199.Google Scholar
[6]Lennox, J. C. and Roseblade, J. E., ‘Centrality in finitely generated soluble groups’, J. Algebra 16 (1970), 399–435.CrossRefGoogle Scholar
[7]Lennox, J. C. and Wiegold, James, ‘Extensions of a problem of Paul Erdos on groups’, J. Austral. Math. Soc. 31 (1981), 459–463.Google Scholar
[8]Napolitani, F., ‘Sui p-gruppi modulari finiti’, Rend. Sem. Mat. Univ. Padova 39 (1967), 296–303.Google Scholar
[9]Neumann, B. H., ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. 21 (1976), 467–472.Google Scholar