Indecomposable solutions of the Yang–Baxter equation of square-free cardinality

Indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation of cardinality , for different prime numbers , are studied. It is proved that they are multipermutation solutions of level ≤n. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [11] and provides a far reaching extension of several earlier results on indecomposability of solutions. The proofs are based on a detailed study of the brace structure on the permutation group associated to such a solution. It is proved that are the only primes dividing the order of . Moreover, the Sylow -subgroups of are elementary abelian -groups and if denotes the Sylow -subgroup of the additive group of the left brace , then there exists a permutation such that , are ideals of the left brace and . In addition, indecomposable solutions of cardinality that are multipermutation of level n are constructed, for every nonnegative integer n.