# Showing Groups of Homomorphisms are Isomorphic

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.

• By "${\rm Hom}(B \times C)$", do you mean "${\rm Hom}(B, C)$"? Feb 27 at 18:12
• Yes, thank you @Shaun I have corrected the question. Feb 27 at 20:03
• You're welcome. See this duplicate. Feb 27 at 20:09

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 $$\operatorname{Hom}(A\times B,C)$$?