The exercise is prove every abelian group $A$ is isomorphic to a factor group of a free abelian group. This was a recommended exercise in Lang's Algebra book (page 39, if you are curious). What I said was choose a set $S$ such that we have a surjective mapping of $g: S \rightarrow A.$ We then have a homomorphism $g_*: Z\langle S\rangle \rightarrow A$ with the mapping $\phi = \sum_{x \in S} k_x \cdot x \mapsto \sum_{x \in S} k_xg(x).$ Since $g$ was surjective, clearly $g_*$ is surjective as well. Hence, we have an isomorphism $Z\langle S\rangle/ker(g_*) \cong A.$
Does this work?