Group ring of products

Question

Let G be a group and denote by $k[G]$ the group ring over a commutative ring $k$. Is then $k[G^n]\cong k[G]^{\otimes n}$? If so, what is the isomorphism?

Thanks a lot!

Answer 1

Yes. This follows from the fact that

$$k[G \times H] \ni \sum c_{ij} (g_i \times h_j) \mapsto \sum c_{ij} g_i \otimes h_j \in k[G] \otimes_k k[H]$$

is an isomorphism.