Some properties of nilpotent groups
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 propert...
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| Datum: | 2018 |
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| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/797 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | 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 property 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 \(G_{0}^{\ast}\) is any homomorphic image of \(G_{0}\), then the set of torsion elements in \(G_{0}^{\ast}\) 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. |
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