Let $G_1. ..., G_k$ be any groups and $\sigma \in S_k$ a permutation. Prove that the function $$ \varphi: G_1 \times \cdots \times G_k \rightarrow G_{\sigma(1)} \times \cdots \times G_{\sigma(k)} $$ $$ \varphi: \hspace{0.2cm} (g_1, \cdots , g_k) \hspace{0.2cm} \mapsto \hspace{0.5cm}(g_{\sigma(1)}, \cdots ,g_{\sigma(k)}) $$ is a homomorphism.
Progress: Let $(x_1,...,x_k), (y_1,...,y_k)\in G_1\times...\times G_k$ then $$ \varphi((x_1,...,x_k)(y_1,...,y_k))=\varphi(x_1 y_1,...,x_k y_k)=???=(x_{\sigma(1)}, \cdots ,x_{\sigma(k)})(y_{\sigma(1)}, \cdots, y_{\sigma(k)})$$ How to fill in the ??? step?
It is probably clear that $\varphi: G_1 \times G_2 \times G_3 \times\cdots \times G_k \rightarrow G_2 \times G_1 \times G_3 \times \cdots \times G_k $ given by $\varphi(g_1,g_2,g_3,\dots,g_k)=(g_2,g_1,g_3,\dots,g_k)$ is an isomorphism.
It is also clear that $1$ and $2$ are not special and can be replaced by any pair $i$ and $j$ with $i\ne j$.
Now use that every permutation in $S_k$ is a product of transpositions to prove the general case.