Theorem 15.16

Let $\phi : G \rightarrow G'$ be a group homomorphism. If $N$ is a normal subgroup of $G$, then $\phi [N]$ is a normal subgroup of $\phi [G]$. Also, if $N'$ is a normal subgroup of $\phi [G]$, then $\phi ^{-1} [N']$ is a normal subgroup of $G$.