Essential subgroups and essential extensions

TL;DR

This paper extends the concepts of essential submodules and extensions from modules to groups, providing characterizations, properties, and a new proof of a classical theorem in group theory.

Contribution

It introduces the extension of essential subgroup concepts to nonabelian groups and characterizes groups lacking proper essential extensions, also offering a new proof of a key theorem.

Findings

Characterization of groups without proper essential extensions

Conditions for a group to have a proper essential subgroup

A short proof of the theorem that only the trivial group is injective in the category of groups

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.