Chris Leary asked me yesterday that: $A=Z_n$ is torsion and $C=Z$ is tosion free. What can you say about $Hom(A,C)$ in this case? With the same C and A=Q, a divisible abelian group, what can you say about Hom(A,C)?
How can you show me an homomorphism non-trival from $Z_n$ to $Z$? suggest that you contruct as:
$f(x+nZ)=x$. clearly $f$ is not map!
i really hope your answer!