# Definition:Maximal Normal Subgroup

## Definition

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, if there is no normal subgroup of $G$, except $N$ and $G$ itself, which contains $N$.