Normal subgroup

Definition

A subgroup $H$ of a group $G$ is termed normal if it satisfies the following equivalent conditions:

• $H$ is the kernel of a homomorphism from $G$, i.e. there is a homomorphism $\varphi :G\to K$ of groups such that ${\varphi }^{-1}(e)=H$
• $xH{x}^{-1}\subseteq H$, or in other words, $xh{x}^{-1}\in H$ for all $x\in G,h\in H$
• $xH{x}^{-1}=H$