Limit Points of Commuting Probabilities of Finite Groups

Thomas Browning

Published 2022-01-24, updated 2022-03-26Version 2

The commuting probability of a finite group $G$ is the probability that two randomly chosen elements commute. Let $S\subseteq(0,1]$ denote the set of all possible commuting probabilities of finite groups. We prove that $S\cup\{0\}$ is closed, which was conjectured by Keith Joseph in 1977.