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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.

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

1 Answer 1

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

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