If $N$ is any normal subgroup of $G$, then the factor group $G/N$ is abelian if and only if $G' \subseteq N$.
In the proof I don't understand why $G/N$ is the homomorphic image of $G/G'$
$G\subseteq N \subseteq G'$
What is the explicit homomorphism? Does homomorphic image mean $G/N \cong G/G'$?