Geometric approach to the modular isomorphism problem: groups of order 64

TL;DR

The paper presents a geometric method using computational algebraic geometry to solve the modular isomorphism problem for groups of order 64, establishing when algebra isomorphisms imply group isomorphisms.

Contribution

It introduces a novel geometric approach to determine group isomorphism from algebra isomorphism in the modular setting for groups of order dividing 64.

Findings

Algebra isomorphism implies group isomorphism for groups of order dividing 64 under certain ring conditions.

The method applies to commutative rings where 2 is not invertible.

Provides a computational algebraic geometry procedure for the problem.

Abstract

We introduce a procedure based on computational algebraic geometry to determine whether two algebras are isomorphic. We then apply it to show that if $R$ is a commutative unital ring in which $2$ is not invertible, $G$ is a group of order dividing $64$ and $H$ some group, then an isomorphism of unital algebras $RG \cong RH$ implies an isomorphism of groups $G \cong H$.