The Modular Isomorphism Problem over all fields

TL;DR

This paper investigates the Modular Isomorphism Problem over various fields, analyzing when algebra isomorphisms imply group isomorphisms, and extends known results from prime fields to general fields of characteristic p.

Contribution

It generalizes positive results of the Modular Isomorphism Problem from prime fields to arbitrary fields of characteristic p, clarifying differences based on the field type.

Findings

Extended positive results from prime fields to general fields of characteristic p

Identified key differences in the problem depending on the field assumed

Provided a revised understanding of the literature on the Modular Isomorphism Problem

Abstract

The Modular Isomorphism Problem asks, if an isomorphism between modular group algebras of finite $p$-groups over a field $F$ implies an isomorphism of the group bases. We explore the differences of knowledge on the problem when $F$ is either assumed to be a prime field or a general field of characteristic $p$. After revising the literature and explaining reasons for the differences, we generalize some of the positive answers to the problem from the prime field case to the general case.