# 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. Note that a maximal subgroup of $G$ is not maximal (http://planetmath.org/MaximalElement) 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. 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 (http://planetmath.org/QuotientGroup) $G/N$ is simple (http://planetmath.org/Simple).

 Title maximal subgroup Canonical name MaximalSubgroup Date of creation 2013-03-22 12:23:46 Last modified on 2013-03-22 12:23:46 Owner yark (2760) Last modified by yark (2760) Numerical id 15 Author yark (2760) Entry type Definition Classification msc 20E28 Synonym maximal proper subgroup Related topic MaximalElement Defines maximal Defines maximal normal subgroup Defines maximal proper normal subgroup Defines simplicity of quotient group