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Soluble groups isomorphic to their non-nilpotent subgroups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Howard Smith  and
James Wiegold
Howard Smith
Affiliation:
Bucknell UniversityLewisburg PA 17837USA e-mail: howsmith@bucknell.edu
James Wiegold
Affiliation:
School of Mathematics Cardiff UniversitySenghennydd Road Cardiff CF24 4YN Wales e-mail: wiegoldj@cardiff.ac.uk
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Abstract

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A group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

MSC classification

Secondary: 20F19: Generalizations of solvable and nilpotent groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 3 , December 1999 , pp. 399 - 411
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Fuchs, L., Infinite Abelian groups (Academic Press, New York, 1970).Google Scholar
[2]Hall, P., Nilpotent groups. Collected works of Philip Hall (Clarendon Press, Oxford, 1988).Google Scholar
[3]Heineken, H. and Mohamed, I. J., ‘A group with trivial centre satisfying the normalizer condition’, J. Algebra 10 (1968), 368–376.CrossRefGoogle Scholar
[4]Newman, M. F. and Wiegold, J., ‘Groups with many nilpotent subgroups’, Arch. Math. 15 (1964), 241–250.CrossRefGoogle Scholar
[5]Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, New York, 1972).CrossRefGoogle Scholar
[6]Smith, H., ‘Groups with few non-nilpotent subgroups’, Glasgow Math. J. 39 (1997), 141–151.CrossRefGoogle Scholar
[7]Smith, H. and Wiegold, J., ‘Groups which are isomorphic to their non-Abelian subgroups’, Rend. Sem. Mat. Univ. Padova 97 (1997), 7–16.Google Scholar
[8]Smith, H. and Wiegold, J., ‘Groups isomorphic to their non-nilpotent subgroups’, Glasgow Math. J. 40 (1998), 257–262.CrossRefGoogle Scholar