# The isomorphism problem for group algebras: a criterion

## TL;DR

This paper introduces hereditary groups over a finite commutative ring and demonstrates that for such groups, algebra isomorphisms of their group algebras imply group isomorphisms, with applications to the modular isomorphism problem for specific p-groups.

## Contribution

The paper defines hereditary groups over a ring and proves that algebra isomorphisms imply group isomorphisms for these groups, advancing the understanding of the isomorphism problem.

## Key findings

- Hereditary groups over a ring are introduced.
- Algebra isomorphisms imply group isomorphisms for hereditary groups.
- New proofs for theorems on p-groups by Deskins and Passi-Sehgal.

## Abstract

Let $R$ be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over $R$. Our main result states that if $G$ is a hereditary group over $R$ then a unital algebra isomorphism between group algebras $RG \cong RH$ implies a group isomorphism $G \cong H$ for every finite group $H$. As application, we study the modular isomorphism problem, which is the isomorphism problem for finite $p$-groups over $R = \mathbb{F}_p$ where $\mathbb{F}_p$ is the field of $p$ elements. We prove that a finite $p$-group $G$ is a hereditary group over $\mathbb{F}_p$ provided $G$ is abelian, $G$ is of class two and exponent $p$ or $G$ is of class two and exponent four. These yield new proofs for the theorems by Deskins and Passi-Sehgal.