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 $aH{a}^{-1}=H$ 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 $H\mathrm{⊴}G$ or $H◁G$ is often used to denote that $H$ is a normal subgroup of $G$.

The kernel $\ker (f)$ of any group homomorphism $f:G⟶G^{\prime }$ 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 $\rho :G⟶G/H$, where $G/H$ is the quotient group).

Title	normal subgroup
Canonical name	NormalSubgroup
Date of creation	2013-03-22 12:08:07
Last modified on	2013-03-22 12:08:07
Owner	djao (24)
Last modified by	djao (24)
Numerical id	11
Author	djao (24)
Entry type	Definition
Classification	msc 20A05
Synonym	normal
Related topic	QuotientGroup
Related topic	Normalizer
Defines	normality