Commuting probabilities of finite groups
Group Theory
2017-02-14 v3 Number Theory
Abstract
The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups.
Cite
@article{arxiv.1411.0848,
title = {Commuting probabilities of finite groups},
author = {Sean Eberhard},
journal= {arXiv preprint arXiv:1411.0848},
year = {2017}
}
Comments
13 pages. To appear in the Bulletin of the LMS. This is the version accepted for publication, incorporating the referee's suggestions. This version will differ from the published version