The isomorphism problem for rational group algebras of finite metacyclic nilpotent groups

TL;DR

This paper proves that for finite metacyclic groups, an isomorphism of their rational group algebras implies the groups themselves are isomorphic, specifically when one is nilpotent.

Contribution

It establishes a new isomorphism criterion for finite metacyclic groups based on their rational group algebras, focusing on the nilpotent case.

Findings

Isomorphic rational group algebras imply group isomorphism for finite metacyclic nilpotent groups.

The result applies specifically when one of the groups is nilpotent.

Provides a new tool for classifying finite metacyclic groups via algebraic invariants.

Abstract

We prove that if $G$ and $H$ are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent then $G$ and $H$ are isomorphic.