The abelianization of a group $G$ is $G/[G,G]$ the quotient of $G$ by its derived subgroup.

The abelianization of $G$ is the largest abelian quotient of $G$ in the sense that if $N$ is a normal subgroup of $G$ then $G/N$ is abelian if and only if $[G,G]\subseteq N$ In particular, every abelian quotient of $G$ is a homomorphic image of $G/[G,G]$

If $A$ is an abelian group and $\phi\colon G\to A$ is a homomorphism, then there is a unique homomorphism $\psi\colon G/[G,G]\to A$ such that $\psi\circ\pi=\phi$ where $\pi\colon G\to G/[G,G]$ is the canonical projection.