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 permutation group G(X,r) has cardinality which is a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated. It is also shown that if A is a left brace whose cardinality is an odd number and (−a)⋅b=−(a⋅b) 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.