The modular isomorphism problem and abelian direct factors

TL;DR

This paper demonstrates that the structure of certain abelian and non-abelian components of a finite p-group can be uniquely determined from its group algebra over a finite field, extending previous results.

Contribution

It generalizes the modular isomorphism problem by showing the group algebra determines key direct factors of p-groups over any field of characteristic p.

Findings

The isomorphism type of the maximal abelian direct factor is determined by the group algebra.

The isomorphism type of the non-abelian remaining factor's group algebra is also determined.

The paper extends known results to arbitrary fields of characteristic p.

Abstract

Let $p$ be a prime and let $G$ be a finite $p$-group. We show that the isomorphism type of the maximal abelian direct factor of $G$, as well as the isomorphism type of the group algebra over $\mathbb F_p$ of the non-abelian remaining direct factor, are determined by $\mathbb F_p G$, generalizing the main result in arXiv:2110.10025 over the prime field. In order to do this, we address the problem of finding characteristic subgroups of $G$ such that their relative augmentation ideals depend only on the $k$-algebra structure of $kG$, where $k$ is any field of characteristic $p$, and relate it to the modular isomorphism problem, reproving and extending some known results.