A note on set-theoretic solutions of the Yang-Baxter equation

This paper shows that every finite non-degenerate involutive set theoretic solution (X,r) of the Yang-Baxter equation whose symmetric group has cardinality which a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated (Theorems 3, 5 and 11). It is also shown that if A is a left brace whose cardinality is an odd number and (-a) b=-(ab) for all a, b A, then A is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level.