The isomorphism problem for group algebras: A criterion

Abstract

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 R ⁢ G ≅ R ⁢ H {RG\cong RH} implies a group isomorphism G ≅ H {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 = 𝔽 p {R=\mathbb{F}_{p}} , where 𝔽 p {\mathbb{F}_{p}} is the field of p elements. We prove that a finite p-group G is a hereditary group over 𝔽 p {\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.