Tensor product of group algebras

by takrp   Last Updated August 25, 2019 05:20 AM

Let $G,G_1$ and $G_2$ are three abelian groups with group homomorphisms $\phi_i:G\to G_i$. This gives $k$-algebra homomorphisms $k[\phi_i]:k[G]\to k[G_i]$. So we can consider $k[G_i]'s$ as $k[G]$-module via the homomorphisms $k[\phi_i]$. We can consider the tensor product $k[G_1]\otimes_{k[G]}k[G_2]$ and this will be again $k$-algebras.

My question is there a simpler way to describe the $k$-algebra: $k[G_1]\otimes_{k[G]}k[G_2]$?

For example take $G=\{e\}$, the identity group; then $k[G]=k$ and hence $$k[G_1]\otimes_{k[G]}k[G_2]=k[G_1]\otimes_kk[G_2]\cong k[G_1\times G_2].$$

So I was wondering if there exists any simpler way to express $k[G_1]\otimes_{k[G]}k[G_2]$ like above.

Note that here the groups are abelian and hence the group algebras are commutative rings. Therefore the tensor product makes sense.

Thank you in advance.