  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.
