Commuting probabilities of finite groups

TL;DR

This paper investigates the set of commuting probabilities in finite groups, proving they are closely related to Egyptian fractions and confirming conjectures about their rationality and order structure.

Contribution

It establishes that all commuting probabilities are nearly Egyptian fractions of bounded complexity and proves two longstanding conjectures about their rationality and ordering.

Findings

All limit points of P are rational.

P is well ordered by >.

Commuting probabilities relate closely to Egyptian fractions.

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.