The kernel of what morphism is the commutator?

Question

The kernel of a morphism is always normal to the group by the first isomorphism theorem. The commutator of a group is normal. Therefore if $G$ is my group and $C(G)$ is its commutators (and their products) is there some morphism $\phi$ such that $Kern(\phi) = C(G)$?

Answer 1

Yes - in general, there's a simple recipe for finding a homomorphism from $G$ with kernel equal to $N$, whenever $N$ is a normal subgroup of $G$. Presumably you know how to form the quotient group $G/N$; can you think of any homomorphisms from $G$ to $G/N$ that might have kernel $N$?