Let be a field or and two groups. It is clear that if then as algebras. **The group ring isomorphism problem** is this question that whether or not as algebras, implies . Another version of the group ring isomorphism problem is this: given a group and a field find all groups such that as algebras. These questions have been answered in special cases only. For example, an old result due to Perlis and Walker states that if are finite abelian groups and as algebras, then In 2001 Hertweek found two non-isomorphic groups of order such that

For now, we’ll only show that it is possible to have and :

**Theorem**. If are two finite abelian groups of order then as -algebras.

*Proof*. We have already seen in this post that for any group So if is a finite abelian group of order then is a commutative semisimple algebra. Thus, since is algebraically closed, the Wedderburn-Artin theorem gives us as -algebras.

**Example**. Let be the Klein four-group and let be the cyclic group of order four. Then but, by the theorem, as -algebras.