The Modular Isomorphism Problem -- the alternative perspective on counterexamples

TL;DR

This paper offers a new perspective on known counterexamples to the Modular Isomorphism Problem, showing they are part of a broader construction and demonstrating limitations for primes greater than 2.

Contribution

It reveals that existing counterexamples are special cases of a more general construction and shows such constructions do not extend to primes larger than 2.

Findings

Counterexamples are special cases of a broader construction

Construction does not produce counterexamples for primes greater than 2

Provides new insights into the structure of the Modular Isomorphism Problem

Abstract

As a result of impressive research arXiv:2106.07231, D. Garc\'{\i}a-Lucas, \'{A}. del R\'{i}o and L. Margolis defined an infinite series of non-isomorphic $2$-groups $G$ and $H$, whose group algebras $\mathbb{F}G$ and $\mathbb{F}H$ over the field $\mathbb{F}=\mathbb{F}_2$ are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for $p>2$ does not provide a similar counterexample to the MIP.