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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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/797 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543070747623425 |
|---|---|
| author | Gaglione, Anthony M. Lipschutz, Seymour Spellman, Dennis |
| author_facet | Gaglione, Anthony M. Lipschutz, Seymour Spellman, Dennis |
| author_sort | Gaglione, Anthony M. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T08:49:22Z |
| description | 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. |
| first_indexed | 2025-12-02T15:50:23Z |
| format | Article |
| id | admjournalluguniveduua-article-797 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:50:23Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-7972018-04-04T08:49:22Z Some properties of nilpotent groups Gaglione, Anthony M. Lipschutz, Seymour Spellman, Dennis Property S, Property R, commensurable, variety of groups, closure operator 20F18,20F05,20F24,16D10 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. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/797 Algebra and Discrete Mathematics; Vol 8, No 4 (2009) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/797/327 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Property S Property R commensurable variety of groups closure operator 20F18,20F05,20F24,16D10 Gaglione, Anthony M. Lipschutz, Seymour Spellman, Dennis Some properties of nilpotent groups |
| title | Some properties of nilpotent groups |
| title_full | Some properties of nilpotent groups |
| title_fullStr | Some properties of nilpotent groups |
| title_full_unstemmed | Some properties of nilpotent groups |
| title_short | Some properties of nilpotent groups |
| title_sort | some properties of nilpotent groups |
| topic | Property S Property R commensurable variety of groups closure operator 20F18,20F05,20F24,16D10 |
| topic_facet | Property S Property R commensurable variety of groups closure operator 20F18,20F05,20F24,16D10 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/797 |
| work_keys_str_mv | AT gaglioneanthonym somepropertiesofnilpotentgroups AT lipschutzseymour somepropertiesofnilpotentgroups AT spellmandennis somepropertiesofnilpotentgroups |