On TFU extensions in LCA groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866907817132163072 |
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| author | Alijani, Aliakbar |
| author_facet | Alijani, Aliakbar |
| contents | Let $\ell$ be the category of all locally compact abelian (LCA) groups. Let $G\in\ell$ and $H\subseteq G$. The first Ulm subgroup of $G$ is denoted by $G^{(1)}$ and the closure of $H$ by $\overline{H}$. A proper short exact sequence $0\to A\stackrelϕ{\to} B\stackrelψ{\to} C\to 0$ in $\ell$ is said to be a $TFU$ extension if $0\to \overline{A^{(1)}}\stackrel{\overlineϕ}{\to} \overline{B^{(1)}}\stackrel{\overlineψ}{\to} \overline{C^{(1)}}\to 0$ is a proper short exact sequence where $\overlineϕ=ϕ\mid_{\overline{A^{(1)}}}$ and $\overlineψ=ψ\mid_{\overline{B^{(1)}}}$. We introduce some results on $TFU$ extensions. Also, we establish conditions under which the $TFU$ extensions split. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16526 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On TFU extensions in LCA groups Alijani, Aliakbar Group Theory General Topology 20K35, 22B05 Let $\ell$ be the category of all locally compact abelian (LCA) groups. Let $G\in\ell$ and $H\subseteq G$. The first Ulm subgroup of $G$ is denoted by $G^{(1)}$ and the closure of $H$ by $\overline{H}$. A proper short exact sequence $0\to A\stackrelϕ{\to} B\stackrelψ{\to} C\to 0$ in $\ell$ is said to be a $TFU$ extension if $0\to \overline{A^{(1)}}\stackrel{\overlineϕ}{\to} \overline{B^{(1)}}\stackrel{\overlineψ}{\to} \overline{C^{(1)}}\to 0$ is a proper short exact sequence where $\overlineϕ=ϕ\mid_{\overline{A^{(1)}}}$ and $\overlineψ=ψ\mid_{\overline{B^{(1)}}}$. We introduce some results on $TFU$ extensions. Also, we establish conditions under which the $TFU$ extensions split. |
| title | On TFU extensions in LCA groups |
| topic | Group Theory General Topology 20K35, 22B05 |
| url | https://arxiv.org/abs/2210.16526 |