In my textbook I found the statement:
It is easy to prove that given $\varphi :S_4 \times \mathbb{Z}_3 \longrightarrow \mathbb{Z}_6$ homomorphism, then $\varphi =\alpha \times \beta$ with $\alpha :S_4 \longrightarrow \mathbb{Z}_6$ homomorphism and $\beta :\mathbb{Z}_3 \longrightarrow \mathbb{Z}_6$ homomorphism, such that $(\alpha \times \beta )(h,k)=\alpha (h)+\beta (k)$
My question is: how can I prove that? And is it possible to generalise the statement the following way:
Given $\varphi :G \times K \longrightarrow H$ group homomorphism, then $\varphi =\alpha \times \beta$ with $\alpha :G \longrightarrow H$ and $\beta :K \longrightarrow H$ homomorphisms such that $(\alpha \times \beta)(g,k)=\alpha (g) \cdot _H \beta (k)$
?