The isomorphism problem for rational group algebras of finite metacyclic groups

TL;DR

This paper proves that for finite metacyclic groups, an isomorphism of their rational group algebras implies the groups themselves are isomorphic, clarifying the boundary of the Isomorphism Problem in this context.

Contribution

It establishes that the rational group algebra completely determines finite metacyclic groups up to isomorphism, advancing understanding of the Isomorphism Problem.

Findings

Rational group algebra isomorphism implies group isomorphism for finite metacyclic groups

Clarifies the boundary between positive and negative solutions to the Isomorphism Problem

Provides a complete characterization for this class of groups

Abstract

We prove that if two finite metacyclic groups have isomorphic rational group algebras, then they are isomorphic. This contributes to understand where is the line separating positive and negative solutions to the Isomorphism Problem for group algebras.