## You are here

Homenormal subgroup

## Primary tabs

# normal subgroup

A subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a\in G$. Equivalently, $H\subset G$ is normal if and only if $aHa^{{-1}}=H$ for all $a\in G$, i.e., if and only if each conjugacy class of $G$ is either entirely inside $H$ or entirely outside $H$.

The notation $H\trianglelefteq G$ or $H\triangleleft G$ is often used to denote that $H$ is a normal subgroup of $G$.

The kernel $\ker(f)$ of any group homomorphism $f:G\longrightarrow G^{{\prime}}$ is a normal subgroup of $G$. More surprisingly, the converse is also true: any normal subgroup $H\subset G$ is the kernel of some homomorphism (one of these being the projection map $\rho:G\longrightarrow G/H$, where $G/H$ is the quotient group).

Defines:

normality

Related:

QuotientGroup, Normalizer

Synonym:

normal

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

20A05*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
- Corrections