Work

Smooth cuboids in group theory

Year: 2025

Type: article

Abstract: A smooth cuboid can be identified with a $3\times 3$ matrix of linear forms, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over $K$. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also give a description of the automorphism group. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite $p$-groups of class $2$ and exponent $p$ arising in this way.

Source: Algebra & Number Theory

DOI: 10.2140/ant.2025.19.967

Authors: Joshua Maglione, Mima Stanojkovski

Institutions: Ollscoil na Gaillimhe – University of Galway, University of Trento

Language: English

Topic: Finite Group Theory Research

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: Deutsche Forschungsgemeinschaft, Daimler und Benz Stiftung

Awards: G2063568005, G251152387, TRR 195: Symbolische Werkzeuge in der Mathematik und ihre Anwendung, Ordnungszetafunktionen von Ganzzahlringen und Auflösung von Singularitäten