On the Modular Isomorphism Problem for groups of class 3 and obelisks

TL;DR

This paper advances understanding of the Modular Isomorphism Problem for class 3 groups by combining existing and novel techniques, including the use of small group algebras and properties of p-obelisks, to identify cases of isomorphism.

Contribution

It introduces a new approach to analyze the lower central series from the modular group algebra and provides positive results for specific classes of nilpotent groups of class 3.

Findings

Positive results for two classes of groups of nilpotency class 3

New method to derive properties of the lower central series

Analysis of p-obelisks using computer-aided investigations

Abstract

We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite $p$-group from the structure of the associated modular group algebra. Finally, we study the class of so-called $p$-obelisks which are highlighted by recent computer-aided investigations of the problem.