Let $G$ be a group. Let $N$ $\trianglelefteq$ $G$.
We define $\phi$ : $G$ $\rightarrow$ $G^{ab}$ a homomorphism, where $G^{ab} : = G/[G,G]$ is the abelianization of group $G$.
Is it true that $G^{ab}/\phi(N) \cong (G/N)^{ab}$ and how do we prove it?
Any help or hint is appreciated.
Edit : My main idea is to find a homomorphism $f$, then find Ker($f$) and Im($f$). First, I tried defining $f : G^{ab} \rightarrow G \rightarrow G/N \rightarrow (G/N)^{ab}$. But it seems like the first arrow is not well defined since $\phi^{-1}$ is not guaranteed. Then I posed $\psi : G^{ab} \rightarrow (G/N)^{ab}$ and wasn’t sure anymore how to continue.