normal subgroup NormalSubgroup 2002-01-05 02:03:30 2007-07-04 05:40:57 Definition normality \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A subgroup $H$ of a group $G$ is {\em 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 $f: G \longrightarrow G'$ 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).