Factorized Groups, Yang-Baxter Equation and local Nearrings

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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.