\to A$ from $G$ into an abelian group has the property that $\text{ker}(\phi)$ contains $[G, G]$. In fact, the quotient $G \to G/[G, G]$ is universal with this property; that is, any other homomorphism $\phi : G \to A$ from $G$ into an abelian group factors through a $G/[G, G] \to