Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T17:52:26.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsion-free groups isomorphic to all of their non-nilpotent subgroups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Patrizia Longobardi ,
Mercede Maj ,
Howard Smith  and
James Wiegold
Patrizia Longobardi
Affiliation:
Dipartimento di Matematica e Informatica, via S. Allende, 84081 Baronissi, Italy e-mail: [email protected] e-mail: [email protected]
Mercede Maj
Affiliation:
Bucknell University, Lewisburg PA 17837, USA e-mail: [email protected]
Howard Smith
Affiliation:
School of Mathematics, Cardiff University, Cardiff CF24 4Y, Wales e-mail: [email protected]
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 result is that every torsion-free locally nilpotent group that is isomorphic to each of its nonnilpotent subgroups is nilpotent, that is, a torsion-free locally nilpotent group G that is not nilpotent has a non-nilpotent subgroup H that is not isomorphic to G.

Keywords

torsion-freelocally nilpotent groups

MSC classification

Secondary: 20F19: Generalizations of solvable and nilpotent groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 3 , December 2001 , pp. 339 - 348
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Hall, P., Nilpotent groups, Collected works of Philip Hall (Clarendon Press, Oxford, 1988).Google Scholar
[2]Möhres, W., ‘Torsionsfreie Gruppen, deren Untergruppen alle subnormal sind’, Math. Ann. 284 (1989), 245–249.CrossRefGoogle Scholar
[3]Newman, M. F. and Wiegold, J., ‘Groups with many nilpotent subgroups’, Arch. Math. 15 (1964), 241–250.CrossRefGoogle Scholar
[4]Ol'shanskii, A. Yu., Geometry of defining relations in groups (Nauka, Moscow, 1989).Google Scholar
[5]Smith, H., ‘Groups with few non-nilpotent subgroups’, Glasgow Math. J. 39 (1997), 141–151.CrossRefGoogle Scholar
[6]Smith, H. and Wiegold, J., ‘Groups which are isomorphic to their nonabelian subgroups’, Rend. Sem. Mat. Univ. Padova 97 (1997), 7–16.Google Scholar
[7]Smith, H. and Wiegold, J., ‘Groups isomorphic to their non-nilpotent subgroups’, Glasgow Math. J. 40 (1998), 257–262.CrossRefGoogle Scholar
[8]Smith, H. and Wiegold, J., ‘Soluble groups isomorphic to their non-nilpotent subgroups’, J. Austral. Math. Soc. (Series A) 67 (1999), 399–411.CrossRefGoogle Scholar