# 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.

Type of Math Object:
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## Mathematics Subject Classification

20E28 Maximal subgroups

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## Info

Owner: yark
Added: 2002-02-19 - 16:54
Author(s): yark

## Versions

(v15) by yark 2013-03-22
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