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 F C-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 F C-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 ∈ G(X, r). This naturally leads to the study of a brace-theoretic analogue of the class of F C-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.