Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $(B,+,\cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,\cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.

中文翻译：

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每个有限阿贝尔群都是有限简单左括号的可加群的子群

由 Rump 引入的左括号已证明为研究量子杨-巴克斯特方程的集合理论解提供了重要工具。特别是，他们允许构建几个新的解决方案系列。左大括号 $(B,+,\cdot )$ 是由集合 $B$ 上的两个群结构决定的结构：一个阿贝尔群 $(B,+)$ 和一个群 $(B,\cdot)$，满足一定的兼容性条件。本文的主要结果表明，每一个有限阿贝尔群$A$都是一个有限简单左括号$B$的加法群的一个子群，这个群是一个带有阿贝尔Sylow子群的metabelian乘法群。这个结果补充了作者早期关于大量有限简单左括号的意外结果。