Algebra-isomorphic groups

This article defines an equivalence relation on groups, parametrized by a field

Definition

Two groups are said to be algebra-isomorphic with respect to a given field if the group algebras of both these groups over the field, are isomorphic as algebras over the field. Note that if two groups are algebra-isomorphic, then there is a bijection between the irreducible representations of one and the irreducible representations of the other, and hence the degrees of irreducible representations]] over the field are the same for the two groups.

Facts

Further information: Isomorphic group algebra problem

Reducing the non-modular case to the rationals case

Further information: Passmann's theorem

Passmann proved that if the group algebras of $G$ and $H$ over $\mathbb{Q}$ are isomorphic, then the group algebras over any field $K$ whose character does not divide the order of $G$, are isomorphic.

Problem of determining algebra-isomorphic groups

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.

Solution for the Abelian non-modular case

Further information: Perlis-Walker theorem

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.

References

• Abelian group algebras of finite order by Sam Perlis and Gordon L. Walker, Transactions of the American Mathematical Society, Vol. 68, No. 3. (May, 1950), pp. 420-426
• Finite groups with isomorphic group algebras by D. B. Coleman, Transactions of the American Mathematical Society, Vol. 68, No. 3. (May, 1950), pp. 420-426
• Finite Abelian Groups with Isomorphic Group Algebras by W. E. Deskins, Duke Mat. J. 23 (1956), 35–40