Commuting probability in algebraic groups

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.