The Modular Isomorphism Problem and Abelian Direct Factors

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 Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_p$$\\end{document} of the non-abelian remaining direct factor, if existing, are determined by FpG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_p G$$\\end{document}, generalizing the main result in Margolis et al. (Abelian invariants and a reduction theorem for the modular isomorphism problem, Journal of Algebra 636, 533-559 (2023)) over the prime field. 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, extending and reproving some known results.