论文标题
群体理论中平滑的立方体
Smooth cuboids in group theory
论文作者
论文摘要
可以用$ 3 \ times 3 $线性形式的矩阵来识别光滑的立方体，并在字段$ k $中具有系数，其决定符描述了投射平面中平滑的立方体。对于每个这样的矩阵，一个人都可以将组计划与$ k $相关联。我们根据它们的伴随代数产生这些组的同构不变，这也提供了有关其最大Abelian亚组数量的信息。此外，我们从椭圆曲线的同构方面给出了组的同构类型的表征，还描述了自多态群体。最后，我们将结果应用于确定自动形态群体的确定，以及对以这种方式产生的有限$ p $ - 级$ 2 $和指数$ p $的同构测试。
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.
