Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T07:21:11.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of a problem of Paul Erdös on groups

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

John C. Lennox  and
James Wiegold
John C. Lennox
Affiliation:
Department of Mathmatics, University College, Cardiff, Wales
James Wiegold
Affiliation:
Department of Mathmatics, University College, Cardiff, Wales
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main results are as follows. A finitely generated soluble group G is polycyclic if and only if every infinite set of elements of G contains a pair generating a polycyclic subgroup; and the same result with “polycyclic” replaced by “coherent”.

MSC classification

Secondary: 20F16: Solvable groups, supersolvable groups 20E25: Local properties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 4 , December 1981 , pp. 459 - 463
Copyright
Copyright © Australian Mathematical Society 1981

References

Groves, J. R. J. (1978), ‘Soluble groups in which every finitely generated subgroups is finitely presented’, J. Austral. Math. Soc. Ser. A 26, 115–125.CrossRefGoogle Scholar
Hall, P. (1956), ‘Finite by nilpotent groups’, Proc. Cambridge Philos. Soc. 52, 611–616.CrossRefGoogle Scholar
Lennox, J. C. (1973), ‘Bigenetic properties of finitely generated hyper-(abelian-by-finite) groups’, J. Austral. Math. Soc. 16, 309–315.CrossRefGoogle Scholar
L'vov, L. V. and Khukhro, I. K. (1978), Kourouka notebook, p. 73 (Unsolved problems in group theory, Novosibirsk).Google Scholar
Neumann, B. H. (1976), ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. Ser. A 21, 467–472.CrossRefGoogle Scholar
Newman, M. F. (unpublished), ‘A theorem of Goled-Šfarevič and an application in group theory’.Google Scholar
Robinson, D. J. S. (1972), Finiteness conditions and generalized soluble groups, (Springer-Verlag, London-Heidelberg-New York).CrossRefGoogle Scholar
Vaughan-Lee, M. R. and Wiegold, James (1981), ‘Countable locally nilpotent groups of finite exponent with no maximal subgroups’, Bull. London Math. Soc. 13, 45–46.CrossRefGoogle Scholar