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arXiv · 1901.09939

The isomorphism problem for group algebras: a criterion

Taro Sakurai
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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.

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Taro Sakurai. 2019-03-20. The isomorphism problem for group algebras: a criterion. https://doi.org/10.1515/jgth-2019-0071

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