Let $G$ be a nonidentity group with identity element $e$. For each [normal subgroup](/page/Normal%20Subgroup) $N \trianglelefteq G$, let $\pi_N: G \to G/N$ denote the quotient homomorphism given by $\pi_N(g)=gN$. Then $G$ is simple if and only if, for every normal subgroup $N \trianglelefteq G$, either $N=\{e\}$ and $\pi_N: G \to G/N$ is an isomorphism, or $N=G$ and $G/N \cong \{e\}$.