Factorized Groups, Yang-Baxter Equation and local Nearrings

许多作者研究了可表示为两个子群A和B的乘积G = AB的群。一个特殊的作用，在这样的调查发挥三重因式分解群的形式G = AB = AM = BM与子群A，B和正常子群M的G，使A和M之间的交叉点分别。B和M是平凡的。事实证明，如果M是阿贝尔的，那么这些群和所谓的大括号之间存在一一对应。这些都是广义根环的研究中出现的集理论解决方案的量子杨巴克斯特方程。在这种情况下，当子群M是非交换群时，三重分解群可以用某些近似，特别是所谓的局部近似来构造。这意味着许多关于大括号和局部邻域结构的问题可以归结为关于三重分解群结构的问题，反之亦然。我们将研究这种联系的不同方面，此外，考虑某些结构问题的因式分解群G = AB的情况下产生的两个子群A和B有交换子群的小指数。

Groups which can be written as a product G = AB of two subgroups A and B have been studied by many authors. A particular role in such investigations play triply factorized groups of the form G = AB = AM = BM with subgroups A, B and a normal subgroup M of G such that the intersections between A and M resp. B and M are trivial. It turns out that if M is abelian, then there exists a one-to-one correspondence between these groups and so-called braces. These are generalized radical rings which arise in the study of set-theoretical solutions of the quantum Yang-Baxter equation. In the case, when the subgroup M is non-abelian, triply factorized groups can be constructed using some nearrings, especially so-called local nearrings. This means that many problems concerning the structure of braces and local nearrings can be reduced to some questions about the structure of triply factorized groups and vice versa. We will study different aspects of this connection and, in addition, consider certain structural questions about factorized groups G = AB arising in the case when the two subgroups A and B have abelian subgroups of small index.