On the Yang-Baxter equation and left nilpotent left braces

We study non-degenerate involutive set-theoretic solutions (X, r) of the Yang-Baxter equation, we call them solutions. We prove that the structure group G(X, r) of a finite non-trivial solution (X, r) cannot be an Engel group. It is known that the structure group G(X, r) of a finite multipermutation solution (X, r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X, r) which are poly-Z groups but not Engel groups. We find an explicit relation between the multipermutation level of a left brace and the length of the radical chain A((n+1)) = A((n)) * A introduced by Rump. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite left brace, or equivalently into a finite involutive braided group. (C) 2016 Elsevier B.V. All rights reserved.