Primitive set-theoretic solutions of the Yang–Baxter equation

To every involutive non-degenerate set-theoretic solution (X,r) of the Yang–Baxter equation on a finite set X there is a naturally associated finite solvable permutation group G(X,r) acting on X. We prove that every primitive permutation group of this type is of prime order p. Moreover, (X,r) is then a so-called permutation solution determined by a cycle of length p. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.