The isomorphism problem for group algebras: a criterion

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.