Smooth cuboids in group theory

TL;DR

This paper explores the structure and classification of groups associated with smooth cuboids, represented by matrices of linear forms, linking algebraic invariants to elliptic curve isomorphisms and automorphism groups.

Contribution

It introduces new invariants for these groups based on their adjoint algebras and characterizes their isomorphism types via elliptic curve isomorphisms.

Findings

Invariants of groups derived from smooth cuboids are established.

Characterization of group isomorphism types via elliptic curves.

Automorphism groups of these groups are described and applied to finite p-groups.

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.