Relative commuting probability and BFC-type results for finite skew braces

Source abstract

We develop a probabilistic approach to the structure of finite skew left braces, motivated both by classical commuting probability in finite group theory and by the role of skew left braces in the study of set-theoretic solutions of the Yang--Baxter equation. We introduce relative commuting probability for skew left braces and establish analogues of several classical structural results, including relative BFC-type theorems. For natural classes of finite skew left braces, we show that a positive lower bound for the commuting probability forces the existence of a large section which is trivial up to bounded subgroups. We also study the probability that two elements generate a trivial sub-skew brace and a Sylow-local version of commuting probability, obtaining further structural consequences. Finally, we introduce a commuting probability for finite non-degenerate set-theoretic solutions of the Yang--Baxter equation. Under natural hypotheses on the associated structure skew brace $G(X,r)$, a positive lower bound for this probability yields a bound on $|G(X,r):\Soc(G(X,r))|$ depending only on the probability and on $|X|$.