Relative commuting probability and BFC-type results for finite skew braces
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|.
Related
- Coprime automorphisms of profinite groups and the commuting probability of Sylow subgroups
- Determinant characters and multiplicative solubility of finite skew braces
- Bisoluble and Binilpotent Skew Braces of Finite Morley Rank
- Classification of finite left-simple skew braces
- On π-nilpotencies of finite skew braces
BibTeX
@misc{mondal2026relative,
title = {{Relative commuting probability and BFC-type results for finite skew braces}},
author = {Susanta Mondal and Pavel Shumyatsky and Marco Trombetti and Manoj Kumar Yadav},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16269},
eprint = {2609.16269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Relative commuting probability and BFC-type results for finite skew braces
AU - Susanta Mondal
AU - Pavel Shumyatsky
AU - Marco Trombetti
AU - Manoj Kumar Yadav
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16269
UR - https://arxiv.org/abs/2609.16269
AB - 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|.
ER -