# 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).

Title normal subgroup NormalSubgroup 2013-03-22 12:08:07 2013-03-22 12:08:07 djao (24) djao (24) 11 djao (24) Definition msc 20A05 normal QuotientGroup Normalizer normality