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Algebra and Discrete Mathematics, 2009, Issue 4, Pages 66–77
(Mi adm144)
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RESEARCH ARTICLE
Some properties of nilpotent groups
A. M. Gaglionea, S. Lipschutzb, D. Spellmanb a Department of Mathematics U.S. Naval Academy
Annapolis, MD 21402 USA
b partment of Mathematics
Temple University
Philadephia, PA 19122 USA
Abstract:
Property S, a finiteness property which can hold in infinite groups, was introduced by Stallings and others and shown to hold in free groups. In [2] it was shown to hold in nilpotent groups as a consequence of a technical result of Mal'cev. In that paper this technical result was dubbed property R. Hence, more generally, any property R group satisfies propert S. In [7] it was shown that property R implies the following (labeled there weak property R) for a group G:
If $G_{0}$ is any subgroup in $G$ and $G_{0}^{*}$ is any homomorphic image of $G_{0}$, then the set of torsion elements in $G_{0}^{*}$ forms a locally finite subgroup.
It was left as an open question in [7] whether weak property R is equivalent to property R. In this paper we give an explicit counterexample thereby proving that weak property R is strictly weaker than property R.
Keywords:
Property S, Property R, commensurable, variety of groups, closure operator.
Received: 23.05.2009 Revised: 23.05.2009
Citation:
A. M. Gaglione, S. Lipschutz, D. Spellman, “Some properties of nilpotent groups”, Algebra Discrete Math., 2009, no. 4, 66–77
Linking options:
https://www.mathnet.ru/eng/adm144 https://www.mathnet.ru/eng/adm/y2009/i4/p66
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