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arXiv · 2407.19309

Essential subgroups and essential extensions

Sourav Koner·Biswajit Mitra
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Abstract

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.

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Sourav Koner, Biswajit Mitra. 2024-07-27. Essential subgroups and essential extensions. https://arxiv.org/abs/2407.19309

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