Smooth cuboids in group theory
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916699905720320 |
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| author | Maglione, Joshua Stanojkovski, Mima |
| author_facet | Maglione, Joshua Stanojkovski, Mima |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03941 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smooth cuboids in group theory Maglione, Joshua Stanojkovski, Mima Group Theory Algebraic Geometry 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. |
| title | Smooth cuboids in group theory |
| topic | Group Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2212.03941 |