Find an epimorphism in Mod-Z

Question

How can I prove that there exists an epimorphism between a direct sum of Z and the set Hom(A,Q/Z) in Mod-Z ?

Answer 1

$\newcommand{\Hom}{\text{Hom}}$$\newcommand{\Z}{\mathbb{Z}}$$\newcommand{\Q}{\mathbb{Q}}$If $A$ is any abelian group, then $\Hom_\Z(A, \Q/\Z)$ is again an abelian group. Also, an epimorphism of abelian groups is just a surjective map. Now any abelian group $G$ is a quotient of a free abelian group: if $G$ has as generating set $\{g_\lambda \mid \lambda \in \Lambda \}$, then there is a surjection $\bigoplus_{\lambda \in \Lambda} \Z \twoheadrightarrow G$ given by sending $1$ (in the $\lambda$-coordinate) to $g_\lambda$ (alternatively, the set map $\Lambda \to G$, $\lambda \mapsto g_\lambda$ induces a group homomorphism $\Z[\Lambda] \to G$ via universal property of free (abelian) groups, which is a surjection precisely if $\{g_\lambda\}$ generate $G$).