I'm about to solve 3 exercice sierge lang algebra third edition:
Let $G^c$ be the sub-group generated by commutator. Question: Show that $G^c$ is normal sub-group of $G$ and show that every homomorphism of $G $ onto an abelian group $G^\prime $ factor through $G / G^c $.
For the first one it is easy, but for the second this is my attempt:
Let $f:G \rightarrow G $ a group homomorphism and consider the canonical projection $\pi : G \rightarrow G^c $ by the first iso theorem there is a unique $\alpha$ s.t for all $g \in G$, $ \alpha (\pi (g))=f(g)$.