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Groups with many permutable subgroups

  • Mario Curzio (a1), John Lennox (a2), Akbar Rhemtulla (a3) and James Wiegold (a2)

Abstract

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.

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Copyright

References

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[1]Adian, S., The Burnside problem and identities in groups, (Ergeb. Math. Grenzgeb., Springer-Verlag, Berlin, Heidelberg and New York, 1979).
[2]Groves, J. R. J., ‘A conjecture of Lennox and Wiegold concerning supersoluble groups’, J. Austral. Math. Soc. 35 (1983), 218220.
[3]Heineken, Hermann and Lennox, John C., ‘A note on products of abelian groups’, Arch. Math. 41 (1983), 498501.
[4]Îto, N., ‘Über das Produkt von zwei abelschen Gruppen’, Math. Z. 62 (1955), 400401.
[5]Iwasawa, K., ‘Über die endlichen Gruppen und die Verbände ihrer Untergruppen’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 4 (1941), 171199.
[6]Lennox, J. C. and Roseblade, J. E., ‘Centrality in finitely generated soluble groups’, J. Algebra 16 (1970), 399435.
[7]Lennox, J. C. and Wiegold, James, ‘Extensions of a problem of Paul Erdos on groups’, J. Austral. Math. Soc. 31 (1981), 459463.
[8]Napolitani, F., ‘Sui p-gruppi modulari finiti’, Rend. Sem. Mat. Univ. Padova 39 (1967), 296303.
[9]Neumann, B. H., ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. 21 (1976), 467472.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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