The modular isomorphism problem and abelian direct factors

2022/09/29 by Diego García-Lucas, García-Lucas, Diego · 1 citation
Mathematics · Computer Science · #Finite Group Theory Research #Rings, Modules, and Algebras #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2209.15128

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.

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