# Isomorphic group algebra problem

## Statement

### Verbal statement

The Isomorphic group algebra problem asks the following: given a group and a field, what are the algebra-isomorphic groups to the given group with respect to the field? In other words, what are the other groups whose group algebra over the given field is the same as the group algebra of the given group?

### Symbolic statement

Given a group $G$ and a field $k$, the isomorphic group algebra problem asks: what are the groups $H$ for which $k(G) \cong k(H)$ as $k$-algebras.

## Finite case

In 1947, in the Michigan Algebra Conference, R. M. Thrall proposed the following problem: Given a finite group $G$ and a field $k$, find all other finite groups $H$ for which $G$ and $H$ are algebra-isomorphic. This is the isomorphic group algebra problem for finite groups.

### Solution for the Abelian non-modular case

The case where $G$ (and hence necessarily $H$) is an Abelian group, and the characteristic of $k$ does not divide the order of $G$, was considered by Perlis and Walker in their paper On Abelian Group Algebras of Finite Order. They obtained a complete characterization.

### Solution for the Abelian modular case

The case where $G$ is an Abelian group and the characteristic of $k$ divides the order of $G$, was considered and solved by Deskins in his paper Finite Abelian Groups with Isomorphic Group Algebras.

### Solution for the non-Abelian modular case

In the non-Abelian case, partial solutions have been obtained for the case of nilpotent groups (which reduces to the csae of group of prime power order). Work in this direction has been recorded, for instance, in Coleman's Finite Groups with Isomorphic Group Algebras.