Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory. This program recently has been initiated by Bachiller and the authors. In this paper we study the simple finite left braces such that the Sylow subgroups of their multiplicative groups are abelian. We provide several new families of such simple left braces. In particular, they lead to the main, surprising result, that shows that there is an abundance of such simple left braces.

中文翻译：

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大量带有阿贝尔倍增Sylow子群的简单左括号

括号由Rump引入，用于研究Yang-Baxter方程的对合非退化简集理论解。最近发现了一种从所有有限左括号的描述中产生所有此类有限解的构造方法。因此，构造和分类所有简单的左牙套是一个基本问题，因为它们可以被视为一般理论的基础。该程序最近由Bachiller及其作者发起。在本文中，我们研究了简单的有限左括号，以便其乘法组的Sylow子组是abelian。我们提供了一些新的此类左括号的家族。尤其是，它们导致了令人惊讶的主要结果，表明存在大量的这种简单的左牙套。