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If $(X,r)$ is a finite non-degenerate set-theoretic solution of theYang--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 toan $FC$-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 arefinitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. Thisnaturally 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 theirconnections with the solutions of the YBE are described: we prove that theyhave good torsion and radical theories and they behave well with respect tocertain nilpotency concepts and finite generation.

On derived-indecomposable solutions of the Yang--Baxter equation