Factorized Groups, Yang-Baxter Equation and local Nearrings

可表示为A、B两个子群的乘积G=AB的群已被许多作者研究过。在这类研究中，一个特殊的角色是G=AB=AM=BM形式的三因式分解群，G的子群A、B和一个正规子群M使得A和M之间的交集分别为.B和M是微不足道的。证明了如果M是交换的，则这些群与所谓的花括号之间存在一一对应关系。这些是在研究量子杨-巴克斯特方程的集合论解时产生的广义根环。在这种情况下，当子群M是非交换子群时，可以利用一些近环，特别是所谓的局部近环来构造三因式分解群。这意味着许多关于大括号和局部近环的结构问题可以归结为关于三因式分解群的结构的问题，反之亦然。我们将研究这种联系的不同方面，并考虑当两个子群A和B具有小指数的交换子群时所产生的关于因式分解群G=AB的某些结构问题。

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.