An abundance of simple left braces with abelian multiplicative Sylow subgroups

• Ferran Cedó ^[1] ; Eric Jespers ^[2] ; Jan Okniński ^[3]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Vrije Universiteit Brussel

     Vrije Universiteit Brussel

     Arrondissement Brussel-Hoofdstad, Bélgica

     
  3. [3] Warsaw University

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 5, 2020, págs. 1309-1332
• Idioma: inglés
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• Resumen

  • 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.