Essential subgroups and essential extensions
Group Theory
2024-07-30 v1
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.
Cite
@article{arxiv.2407.19309,
title = {Essential subgroups and essential extensions},
author = {Sourav Koner and Biswajit Mitra},
journal= {arXiv preprint arXiv:2407.19309},
year = {2024}
}