Hom, Tensor and Limits

It is trivial (containing only the trivial map $a\mapsto 1_B$) if either $A$ or $B$ is trivial, but otherwise nontrivial.

If $A$ is generated by its torsion elements, then $\operatorname{Hom}(A,B)$ must be trivial.

The group law on $B$ gives $\operatorname{Hom}(A,B)$ a group structure.