Let $G$ be a group with identity element $e_G$, and assume $G \ne \{e_G\}$. Then $G$ is simple if and only if, for every group $H$ and every [group homomorphism](/page/Group%20Homomorphism) $\varphi: G \to H$, one has either $\ker \varphi = \{e_G\}$ or $\ker \varphi = G$.