Commuting probability in algebraic groups

TL;DR

This paper introduces and computes the commuting probability for algebraic groups, extending concepts from finite and compact groups, and explores its behavior across various classes of algebraic groups.

Contribution

It defines commuting probability for algebraic groups, introduces new equivalence notions, and computes this probability for several classes of groups.

Findings

Computed p(G) for reductive, solvable, and nilpotent groups.

Identified limit points of p(G) across different group classes.

Established connections between regular elements and commuting probability.

Abstract

We introduce the notion of commuting probability, $p(G)$, for an algebraic group $G$. This notion is inspired by the corresponding notions in finite groups and compact groups. The computation of $p(G)$ for reductive groups is readily done using the notion of $z$-classes. We introduce two generalisations of this relation, $iz$-equivalence and $dz$-equivalence. These notions lead us naturally to the notion of a regular element in $G$. Finally, with the help of this notion of regular elements, we compute $p(G)$ for a connected, linear algebraic group $G$. We also compute the set of limit points of the numbers $p(G)$ as $G$ varies over the classes of reductive groups, solvable groups and nilpotent groups.