# Definition:Minimal Subgroup

## Definition

Let $G$ be a group.

Let $M \le G$ be a non-trivial subgroup of $G$.

Then $M$ is a minimal subgroup of $G$ if and only if:

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 $M$.