  We study non-degenerate involutive set-theoretic solutions (X,r) of the
Yang-Baxter equation, we call them simply solutions. We show 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 also show that a finite solution of the Yang-Baxter equation can be
embedded in a convenient way into a finite brace and into a finite braided
group.
  For a left brace A, we explore the close relation between the
multipermutation level of the solution associated with it and the radical chain
$A^{(n+1)}=A^{(n)}* A$ introduced by Rump.
