While looking through prior exam problems for a group theory course, I encountered this question and am having some difficulty getting started.
Let $A,B,C$ be abelian groups. Let ${\rm Hom}(A \times B, C)$ be the set of all group homomorphisms from $A \times B$ to $C$. The question asks to show that the group ${\rm Hom}(A \times B, C)$ is isomorphic to ${\rm Hom}(A,C) \times{\rm Hom}(B,C).$
A hint would be appreciated, however, I have generally struggled with proving groups are isomorphic and if there are some general tips/patterns to look for that would be appreciated as well.
Hint: If $\varphi_A\in\operatorname{Hom}(A,C)$ and $\varphi_B\in\operatorname{Hom}(B,C)$, conside the map $\varphi\colon A\times B\longrightarrow C$ defined by $\varphi(a,b)=\varphi_A(a)\varphi_B(b)$. Does it belong to $\operatorname{Hom}(A\times B,C)$?