An abundance of simple left braces with abelian multiplicative Sylow subgroups

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.

Constructing finite simple solutions of the Yang-Baxter equation

“…Of course, nontrivial finite simple left braces B satisfy B = B 2 and Soc(B) = {0}. There are several constructions of finite simple left braces, see for example [13] and the references therein, but we do not know the answer to the following questions.…”

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

“…Example 2.10. There exists a unique simple left brace of size 72, see [12,Remark 4.5] and [38,Proposition 4.3]. The multiplicative group of this left brace is isomorphic to A 4 × S 3 and therefore all of its Sylow subgroups are abelian.…”

“…This rich structure shows that the Yang-Baxter equation is related to different topics such as Hopf-Galois extensions, regular subgroups and nil-rings. For that reason, the theory of (skew) braces is intensively studied, see for example [2,6,8,9,11,12,14,16,20].…”

New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces

“…For basic background and results on involutive non-degenerate set theoretic solutions of the Yang-Baxter equation and on braces we refer to [9] and its references, and [3,6,8,11,12,16,19,18,21,23,24,25,26].…”

“…Our approach is motivated by the decomposition of X into imprimitivity blocks under the action of G(X, r). It also explores and applies nontrivial relations to the class of simple left braces, recently investigated in [1,2,11].…”