  We involve simultaneously the theory of matched pairs of 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
(a braided involutive group) and a left brace, and 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 symmetric 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.
We find new criteria necessary and sufficient to claim that $(X, r)$ is a
multipermutation solution.
