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

Abstract

Indecomposable involutive non-degenerate set-theoretic solutions (X,r) of the Yang–Baxter equation of cardinality p1⋯pn, for different prime numbers p1,…,pn, 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 G(X,r) associated to such a solution. It is proved that p1,…,pn are the only primes dividing the order of G(X,r). Moreover, the Sylow pi-subgroups of G(X,r) are elementary abelian pi-groups and if Pi denotes the Sylow pi-subgroup of the additive group of the left brace G(X,r), then there exists a permutation σ∈Symn such that Pσ(1),Pσ(1)Pσ(2),…, Pσ(1)Pσ(2)⋯Pσ(n) are ideals of the left brace G(X,r) and G(X,r)=P1P2⋯Pn. In addition, indecomposable solutions of cardinality p1⋯pn that are multipermutation of level n are constructed, for every nonnegative integer n.