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maximal subgroup
Let $G$ be a group.
A subgroup $H$ of $G$ is said to be a maximal subgroup of $G$ if $H\neq G$ and there is no subgroup $K$ of $G$ such that $H<K<G$ . Note that a maximal subgroup of $G$ is not maximal among all subgroups of $G$ , but only among all proper subgroups of $G$ . For this reason, maximal subgroups are sometimes called maximal proper subgroups.
Similarly, a normal subgroup $N$ of $G$ is said to be a maximal normal subgroup (or maximal proper normal subgroup) of $G$ if $N\neq G$ and there is no normal subgroup $K$ of $G$ such that $N<K<G$ . We have the following theorem:
Theorem A normal subgroup $N$ of a group $G$ is a maximal normal subgroup if and only if the quotient $G/N$ is simple.
maximal subgroup is owned by yark, Matt Piatkus.
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