\begin{definition}[Definition:Maximal Normal Subgroup]
Let $G$ be a group.
Let $N \le G$ be a proper normal subgroup.
Then $N$ is a '''maximal normal subgroup of $G$''' {{iff}}:
:For every normal subgroup $M$ of $G$, $N \subseteq M \subseteq G$ implies $N = M$ or $M = G$.
That is, {{iff}} there is no normal subgroup of $G$, except $N$ and $G$ itself, which contains $N$.
\end{definition}