Revista Matemática Iberoamericana
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An abundance of simple left braces with abelian multiplicative Sylow subgroupsFerran Cedó, Eric Jespers and Jan Okniński (1) Universitat Autònoma de Barcelona, Bellaterra, Spain
(2) Vrije Universiteit Brussel, Bruxelles, Belgium
(3) Warsaw University, Poland
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.
Keywords: Yang–Baxter equation, set-theoretic solution, brace, simple, $A$-group
Cedó Ferran, Jespers Eric, Okniński Jan: An abundance of simple left braces with abelian multiplicative Sylow subgroups. Rev. Mat. Iberoam. Electronically published on January 16, 2020. doi: 10.4171/rmi/1168 (to appear in print)