The Modular Isomorphism Problem: a survey

TL;DR

This survey reviews the history, strategies, and recent negative resolution of the Modular Isomorphism Problem, which questions whether isomorphic group algebras imply isomorphic finite p-groups.

Contribution

It provides a comprehensive overview of the problem, summarizes the developed strategies, and presents the recent counterexample resolving the long-standing question.

Findings

The problem has a negative solution for certain cases.

Strategies for studying the problem have evolved over time.

Open questions remain in the area of group ring isomorphisms.

Abstract

The Modular Isomorphism Problem asks if an isomorphism of group algebras of two finite p-groups G and H over a field of characteristic p, implies an isomorhism of the groups G and H. We survey the history of the problem, explain strategies which were developed to study it and present the recent negative solution of the problem. The problem is also compared to other isomorphism problems for group rings and various question remaining open are included.