We find an example of a finite solvable group (in fact, a finite p-group) without any left brace structure (equiv. which is not an IYB group). Our argument is an improvement of an argument of Rump, using previous work in other areas of Burde, and of Featherstonhaugh, Caranti and Childs, which we relate to brace theory.

Given a left brace G, a method to construct all the involutive, nondegenerate set-theoretic solutions (Y, s) of the YBE, such that G(Y, s) ∼ = G is given. This method depends entirely on the brace structure of G.

Given a skew left brace B, a method is given to construct all the non-degenerate set-theoretic solutions (X, r) of the Yang Baxter equation such that the associated permutation group G(X, r) is isomorphic, as a skew left brace, to B. This method depends entirely on the brace structure of B. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals except the trivial and the total ideal, called simple left braces. In this article, we present the first non-trivial examples of finite simple left braces. To explicitly construct them, we define the matched product of two left braces, which is a useful method to recover a finite left brace from its Sylow subgroups.

Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace B, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to B. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to construct new families of finite simple left braces.