Commuting probability of skew left braces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911523375415296 |
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| author | Mondal, Susanta Yadav, Manoj K. |
| author_facet | Mondal, Susanta Yadav, Manoj K. |
| contents | We introduce a concept of the commuting probability of a skew left brace analogous to group theory. We establish upper and lower bounds for the commuting probability and prove that, for finite non-trivial skew left braces, it is always at most $\frac{3}{4}$. Interestingly, there is no skew left brace with commuting probability in the open interval $(5/8, 1)$, except $\frac{3}{4}$, for which we construct an explicit example. A characterization of skew left braces having commuting probability $\frac{3}{4}$ or $\frac{5}{8}$ is presented. We further show that the finite skew left braces with commuting probability larger than $\frac{65}{128}$ are necessarily nilpotent. We prove that the commuting probability remains invariant under isoclinism of skew braces. We introduce a concept of a compact Hausdorff topological skew left brace $B$, where we prove that the set of all elements of $B$ having finite centraliser index in $B$ is a Borel subgroup. For such infinite non-trivial skew left braces too $\frac{3}{4}$ is the upper bound for the commuting probability, and $\frac{3}{4}$ is the only rational number which occurs as commuting probability in the open interval $(5/8, 1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16771 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Commuting probability of skew left braces Mondal, Susanta Yadav, Manoj K. Group Theory Rings and Algebras 16N99, 16T25, 20F24 We introduce a concept of the commuting probability of a skew left brace analogous to group theory. We establish upper and lower bounds for the commuting probability and prove that, for finite non-trivial skew left braces, it is always at most $\frac{3}{4}$. Interestingly, there is no skew left brace with commuting probability in the open interval $(5/8, 1)$, except $\frac{3}{4}$, for which we construct an explicit example. A characterization of skew left braces having commuting probability $\frac{3}{4}$ or $\frac{5}{8}$ is presented. We further show that the finite skew left braces with commuting probability larger than $\frac{65}{128}$ are necessarily nilpotent. We prove that the commuting probability remains invariant under isoclinism of skew braces. We introduce a concept of a compact Hausdorff topological skew left brace $B$, where we prove that the set of all elements of $B$ having finite centraliser index in $B$ is a Borel subgroup. For such infinite non-trivial skew left braces too $\frac{3}{4}$ is the upper bound for the commuting probability, and $\frac{3}{4}$ is the only rational number which occurs as commuting probability in the open interval $(5/8, 1)$. |
| title | Commuting probability of skew left braces |
| topic | Group Theory Rings and Algebras 16N99, 16T25, 20F24 |
| url | https://arxiv.org/abs/2603.16771 |