The Elements with Property (s) of a Skew Brace Need Not Form an Ideal: On a Question of Colazzo, Ferrara and Trombetti

An element \(x\) of a skew brace \(B\) has property (s), in the sense of Colazzo, Ferrara and Trombetti, if it has only finitely many conjugates of each of the kinds \(x*b\), \(b*x\), \(b+x-b\) and \(b\circ x\circ b^{-1}\). In her abstract in Oberwolfach Report 9/2023, Colazzo asks whether the elements with property (s) of a skew brace form an ideal; by a result of the three authors they do so in two-sided braces. We answer the question in the negative. On \(\mathbb{Z}\times\mathbb{Z}/4\), with componentwise addition, the multiplication \((m,a)\circ(n,b)=(m+\varepsilon(a)n,\,a+b+2ab)\), where \(\varepsilon(a)=1\) for \(a\in\{0,1\}\) and \(\varepsilon(a)=-1\) for \(a\in\{2,3\}\), defines a left brace in which the elements with property (s) are those of \(\mathbb{Z}\times\{0,1\}\); this set is not closed under addition and not invariant under the maps \(\lambda_b\). A sub-brace with additive group \(\mathbb{Z}\times\mathbb{Z}/2\) has the same property. The verification is elementary and is given in full. On the positive side, in every skew brace these elements form a normal subgroup of the multiplicative group, so only the additive conditions can fail; and they form an ideal in two-sided skew braces, in braces with torsion-free additive group, in bi-skew braces, and in skew braces whose multiplicative group is an FC-group. In semidirect products of a trivial brace by a finite skew brace \(Q\) the set in question is given by a normal subgroup of the multiplicative group of \(Q\), which can be prescribed; this yields examples for each way in which the additive conditions can fail, among them a skew brace with torsion-free additive group, a centrally nilpotent brace of exponent \(4\), and the structure skew brace of a non-degenerate solution of the Yang–Baxter equation with six elements. The question, the notion and the positive answer for two-sided braces are due to Colazzo, Ferrara and Trombetti; the brace on \(\mathbb{Z}/4\) underlying the example is Example 3.8 of their paper, and the semidirect product of skew braces is a known construction. That the elements with property (s) form a subgroup of the multiplicative group follows at once from a characterisation in a recent preprint of Di Matteo and Ferrara together with a result of Cascella, Properzi and Van Antwerpen; it also follows from a result of Mondal and Yadav on a set which we show to be the same. The statement on two-sided skew braces is a short consequence of the first two results. Our proofs are included for completeness. The example also shows that Lemma 3.12 of the paper of Colazzo, Ferrara and Trombetti needs an additional hypothesis; it holds for skew braces with property (S), where it is used. The literature was searched within the limits stated in the text (one citing paper was seen in abstract only). A search that finds nothing is not a proof of novelty, and no priority is claimed. This is an unrefereed note.