ON DERIVED-INDECOMPOSABLE SOLUTIONS OF THE YANG-BAXTER EQUATION

If ( X, r ) is a finite non-degenerate set-theoretic solution of the Yang-Baxter equation, the additive group of the structure skew brace G(X, r ) is an FC- group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to being an FC- group itself. If one additionally assumes that the derived solution of ( X, r ) is indecomposable, then for every element b of G(X, r ) there are finitely many elements of the form b * c and c * b, with c is an element of G(X, r ). This naturally leads to the study of a brace-theoretic analogue of the class of FC- groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories, and that they behave well with respect to certain nilpotency concepts and finite generation.