## You are here

Homemaximal subgroup

## Primary tabs

# 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:

Defines:

maximal, maximal normal subgroup, maximal proper normal subgroup, simplicity of quotient group

Related:

MaximalElement

Synonym:

maximal proper subgroup

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

20E28*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier