# Normal subgroup

The article on this topic in the Group Properties Wiki can be found at: normal subgroupTemplate:Subgroup property

## 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 $\phi:G \to K$ of groups such that $\phi^{-1}(e) = H$
• $xHx^{-1} \subseteq H$, or in other words, $xhx^{-1} \in H$ for all $x \in G, h \in H$
• $xHx^{-1} = H$