Commuting probability of skew left braces

TL;DR

This paper explores the commuting probability in skew left braces, establishing bounds, characterizing specific values, and analyzing invariance under isoclinism, with implications for finite and topological cases.

Contribution

It introduces the concept of commuting probability for skew left braces, provides bounds, characterizations, and invariance properties, and extends the analysis to topological skew braces.

Findings

Maximum commuting probability is 3/4 for finite non-trivial skew left braces.

No skew left brace has commuting probability in (5/8, 1) except 3/4.

Finite skew left braces with probability > 65/128 are nilpotent.

Abstract

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 $(5/8, 1)$.