Essential subgroups and essential extensions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866907995233845248 |
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| author | Koner, Sourav Mitra, Biswajit |
| author_facet | Koner, Sourav Mitra, Biswajit |
| contents | The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated. Further, we have completely characterized groups that do not possess a proper essential extension. These observations are used in concluding several properties of groups having essential subgroups. Finally, a short proof of the well-known theorem of Eilenberg and Moore that the only injective object in the category of groups is the trivial group is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Essential subgroups and essential extensions Koner, Sourav Mitra, Biswajit Group Theory The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated. Further, we have completely characterized groups that do not possess a proper essential extension. These observations are used in concluding several properties of groups having essential subgroups. Finally, a short proof of the well-known theorem of Eilenberg and Moore that the only injective object in the category of groups is the trivial group is given. |
| title | Essential subgroups and essential extensions |
| topic | Group Theory |
| url | https://arxiv.org/abs/2407.19309 |