maximal subgroup

Let $G$ be a group.

A subgroup $H$ of $G$ is said to be a maximal subgroup of $G$ if $H\ne G$ and there is no subgroup $K$ of $G$ such that . 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 $N\ne G$ and there is no normal subgroup $K$ of $G$ such that . 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) $G\mathrm{/}N$ 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