\begin{definition}[Definition:Minimal Subgroup]
Let $G$ be a group.
Let $M \le G$ be a non-trivial subgroup of $G$.
Then $M$ is a '''minimal subgroup of $G$''' {{iff}}:
:For every subgroup $H$ of $G$, $H \subseteq M$ means $H = M$ or $H = \set e$.
That is, if there is no subgroup of $M$, except $M$ and $\set e$ itself, which is a subset of $M$.
\end{definition}