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Braces were introduced by Rump to study involutive non-degenerateset-theoretic solutions of the Yang-Baxter equation. A constructive method forproducing all such finite solutions from a description of all finite leftbraces has been recently discovered. It is thus a fundamental problem toconstruct and classify all simple left braces, as they can be considered asbuilding blocks for the general theory. This program recently has beeninitiated by Bachiller and the authors. In this paper we study the simplefinite left braces such that the Sylow subgroups of their multiplicative groupsare abelian. We provide several new families of such simple left braces. Inparticular, they lead to the main, surprising result, that shows that there isan abundance of such simple left braces.

An abundance of simple left braces with abelian multiplicative Sylow subgroups