Set-theoretic solutions of the Yang-Baxter equation, braces, and symmetric groups

We involve simultaneously the theory of braided groups and the theory of braces to study set-theoretic solutions of the Yang-Baxter equation (YBE). We show the intimate relation between the notions of "a symmetric group", in the sense of Takeuchi, i.e. "a braided involutive group", and "a left brace". We find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group $(G,r)$, \emph{the derived chain of ideals of} $G$, which gives a precise information about the recursive process of retraction of $G$. We prove that every symmetric group $(G,r)$ of finite multipermutation level $m$ is a solvable group of solvable length at most $m$. To each set-theoretic solution $(X,r)$ of YBE we associate two invariant sequences of involutive braided groups: (i) the sequence of its derived symmetric groups; (ii) the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction $(X,r)$. We find new criteria necessary and sufficient to claim that $(X, r)$ is a multipermutation solution