On the isomorphism problem for unit groups of modular group algebras

TL;DR

This paper demonstrates that for all 2-groups of order up to 32, their normalized unit groups in modular group algebras over GF(2) uniquely determine the group, using computational methods.

Contribution

It provides computational evidence that the isomorphism class of normalized unit groups determines small 2-groups in modular group algebras over GF(2).

Findings

All 2-groups of order ≤ 32 are distinguished by their unit groups.

Computational verification using GAP and LAGUNA supports the isomorphism problem.

The result suggests potential for broader classification in modular algebra contexts.

Abstract

Using the computational algebra system GAP (http://www.gap-system.org) and the GAP package LAGUNA (http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm), we checked that all 2-groups of order not greater than 32 are determined by normalized unit groups of their modular group algebras over the field of two elements.