Smooth cuboids
The preprint "Smooth cuboids in group theory" by Joshua Maglione and Mima Stanojkovski has recently been uploaded to the arXiv!
Link to the paper: arXiv:2212.03941
Abstract:
A smooth cuboid can be identified with a 3×3 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.