Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras

TL;DR

This paper investigates whether non-isomorphic 2-groups with dihedral central quotients can have isomorphic modular group algebras, providing new negative solutions and insights into the algebraic structures involved.

Contribution

It extends previous results by classifying all groups in a specific class of 2-groups regarding their modular group algebras, revealing new negative solutions and algebra isomorphisms.

Findings

Discovered new negative solutions for isomorphic modular group algebras among certain 2-groups.

Identified simple algebra isomorphisms that explain some of the negative solutions.

Provided insights into structural properties that distinguish groups with isomorphic algebras.

Abstract

The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by Garc\'ia-Lucas, Margolis and del R\'io [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these negative solutions in the class of two-generated finite $2$-groups with dihedral central quotient, and solve the original question for all groups within this class. As a result, we discover new negative solutions and simple algebra isomorphisms. At the same time, the positive solutions for most of the groups in this class give some insights what makes the negative solutions special.