An abundance of simple left braces with abelian multiplicative Sylow subgroups

TL;DR

This paper explores the construction and classification of simple finite left braces with abelian Sylow subgroups, revealing a surprising abundance of such structures and advancing the understanding of solutions to the Yang-Baxter equation.

Contribution

The paper introduces new families of simple left braces with abelian Sylow subgroups, demonstrating their abundance and contributing to the classification program.

Findings

Several new families of simple left braces identified

Proves the abundance of such braces

Advances the classification of braces related to Yang-Baxter solutions

Abstract

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.