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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2021
- DOI:
- 10.48550/arXiv.2105.12550
- arXiv:
- arXiv:2105.12550
- Bibcode:
- 2021arXiv210512550G
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Algebraic Geometry;
- 14L;
- 20G