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 $f: G \longrightarrow G'$ 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).