Find the rank of Hom(G,Z)？ （1）
Prove that for any finitely generated abelian group G, the set Hom(G, Z) is a free Abelian group of finite rank.
（2）
Find the rank of Hom(G,Z) if the group G is generated by three generators x, y, z with relations 
2x + 3y + z = 0, 
2y - z = 0

(i) Apply the structure theorem: write $$G \simeq \mathbb{Z}^r \oplus_i \mathbb{Z}/d_i$$
Now from here we compute $$Hom(\mathbb{Z}^r \oplus_i \mathbb{Z}/d_i, \mathbb{Z}) \simeq Hom(\mathbb{Z}^r, \mathbb{Z}) \oplus_i Hom(\mathbb{Z}/d_i, \mathbb{Z}) \simeq \mathbb{Z}^r$$
(ii) We just need to find the rank of the free part of $G$, we have it cut out as the cokernel of the map $\mathbb{Z}^2 \rightarrow \mathbb{Z}^3$, given by $$\begin{pmatrix} 2 & 0 \\ 3 & 2 \\ 1 & -1 \end{pmatrix} \sim \begin{pmatrix} 2 & 2 \\ 3 & 5 \\ 1 & 0 \end{pmatrix} \sim \begin{pmatrix} 0 & 2 \\ 0 & 5 \\ 1 & 0 \end{pmatrix} \sim \begin{pmatrix} 0 & 0\\ 0 & 1 \\ 1 & 0 \end{pmatrix}$$
So if I didn't goof that up, our group is simply $\mathbb{Z}^3/\mathbb{Z}^2 \simeq \mathbb{Z}$, and hence Hom is again $\mathbb{Z}$. 
