Smooth cuboids in group theory

Maglione, Joshua; Stanojkovski, Mima

doi:10.2140/ant.2025.19.967

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Smooth cuboids in group theory

Joshua Maglione and Mima Stanojkovski

Vol. 19 (2025), No. 5, 967–1006

DOI: 10.2140/ant.2025.19.967

Abstract

A smooth cuboid can be identified with a $3 \times 3$ matrix of linear forms in three variables, 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, when $K$ is finite, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also describe their automorphism groups. 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.

Keywords

finite $p$-groups, isomorphism testing, automorphism groups, Baer correspondence, Pfaffians, determinantal representations of cubics, elliptic curves

Mathematical Subject Classification

Primary: 20D15, 14M12

Secondary: 68Q25, 11G20, 15A69, 20D45

Milestones

Received: 17 April 2023

Revised: 10 June 2024

Accepted: 16 July 2024

Published: 22 April 2025

Authors

Joshua Maglione
School of Mathematical and Statistical Sciences University of Galway Galway Ireland

Mima Stanojkovski
Dipartimento di Matematica Università di Trento Trento Italy

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Vol. 19, No. 5, 2025