Let $G$ be a group with identity element $e_G$, and let $N \le G$ be a subgroup. Then $N \trianglelefteq G$ if and only if there exist a group $K$ with identity element $e_K$ and a [group homomorphism](/page/Group%20Homomorphism) $\varphi: G \to K$ such that $N = \ker \varphi = \{g \in G : \varphi(g) = e_K\}$.