A note on co-Hopfian groups and rings
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917339916664832 |
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| author | Gaglione, Anthony M. Spellman, Dennis |
| author_facet | Gaglione, Anthony M. Spellman, Dennis |
| contents | Let $p$ and $n$ be positive integers. Assume additionally that $p\neq 3$ is a prime and that $n>2$. Let $R$ be a field of characteristic $p$. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that $SL_{n}(R)$ is co-Hopfian as a group if and only if $R$ is co-Hopfian as a ring. In this paper, we prove that if $k$ is the algebraic closure of the $2$ element field, then $SL_{2}(k)$ is a co-Hopfian group. Since this $k$ is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on co-Hopfian groups and rings Gaglione, Anthony M. Spellman, Dennis Group Theory Primary 20E26, 03C07, Secondary 20F19, 20F05 Let $p$ and $n$ be positive integers. Assume additionally that $p\neq 3$ is a prime and that $n>2$. Let $R$ be a field of characteristic $p$. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that $SL_{n}(R)$ is co-Hopfian as a group if and only if $R$ is co-Hopfian as a ring. In this paper, we prove that if $k$ is the algebraic closure of the $2$ element field, then $SL_{2}(k)$ is a co-Hopfian group. Since this $k$ is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra). |
| title | A note on co-Hopfian groups and rings |
| topic | Group Theory Primary 20E26, 03C07, Secondary 20F19, 20F05 |
| url | https://arxiv.org/abs/2511.03505 |