Abelian invariants and a reduction theorem for the modular isomorphism problem

TL;DR

This paper advances the understanding of the modular isomorphism problem by identifying abelian factors that can be ignored, introducing new invariants, and applying these findings to specific classes of groups.

Contribution

It introduces a reduction theorem that disregards elementary abelian factors and develops four new series of abelian invariants for the modular group algebra of finite p-groups.

Findings

Elementary abelian factors can be disregarded in the modular isomorphism problem.

Four new series of abelian invariants are established.

Results are applied to new classes of groups.

Abstract

We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the modular group algebra of a finite $p$-group. Finally, we apply our results to new classes of groups.