Limit Points of Commuting Probabilities of Finite Groups

TL;DR

This paper proves that the set of all possible commuting probabilities of finite groups, along with zero, forms a closed set, confirming a conjecture from 1977.

Contribution

It establishes that the set of commuting probabilities of finite groups, plus zero, is closed, resolving a long-standing conjecture.

Findings

The set of commuting probabilities plus zero is closed.

Confirmed Keith Joseph's 1977 conjecture.

Provides a comprehensive understanding of commuting probabilities.

Abstract

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.