# 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.