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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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/154599 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Some properties of nilpotent groups / A.M. Gaglione, S. Lipschutz, D. Spellman // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 66–77. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862746292905574400 |
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| author | Gaglione, A.M. Lipschutz, S. Spellman, D. |
| author_facet | Gaglione, A.M. Lipschutz, S. Spellman, D. |
| citation_txt | Some properties of nilpotent groups / A.M. Gaglione, S. Lipschutz, D. Spellman // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 66–77. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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 G₀ is any subgroup in G and G₀* is any homomorphic image of G₀, then the set of torsion elements in G₀* 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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| first_indexed | 2025-12-07T20:45:01Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-154599 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T20:45:01Z |
| publishDate | 2009 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Gaglione, A.M. Lipschutz, S. Spellman, D. 2019-06-15T16:41:10Z 2019-06-15T16:41:10Z 2009 Some properties of nilpotent groups / A.M. Gaglione, S. Lipschutz, D. Spellman // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 66–77. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20F18,20F05,20F24,16D10. https://nasplib.isofts.kiev.ua/handle/123456789/154599 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 G₀ is any subgroup in G and G₀* is any homomorphic image of G₀, then the set of torsion elements in G₀* 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. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Some properties of nilpotent groups Article published earlier |
| spellingShingle | Some properties of nilpotent groups Gaglione, A.M. Lipschutz, S. Spellman, D. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154599 |
| work_keys_str_mv | AT gaglioneam somepropertiesofnilpotentgroups AT lipschutzs somepropertiesofnilpotentgroups AT spellmand somepropertiesofnilpotentgroups |