# Joseph’s conjectures about commuting probability

The commuting probability $P(G)$ of a finite group $G$ is defined to be the probability that two uniformly random elements $x, y \in G$ commute. If $G$ has order $n$ and $k$ conjugacy classes, this is the same thing as $k / n$. You might think this is a good way of measuring how close a group is to being abelian, but a remarkable and well-known fact is that no finite group satisfies $5/8 < P(G) < 1$, so in fact nonabelian groups never get close to being abelian, in this sense.

In the 1970s Keith Joseph made three insightful conjectures about the set of all possible commuting probabilities $S = \{P(G) : G~\text{a finite group}\}$.

1. every limit point of $S$ is rational,
2. for every $x \in (0, 1]$ there is some interval $(x-\epsilon, x)$ that $S$ avoids,
3. every nonzero limit point of $S$ is in $S$.

Note that 3 is stronger than 1.

In 2014 I proved the first two of these conjectures (building on an earlier paper of Hegarty), and then I publicly expressed doubt about the third. My doubt was largely based on the observation that there is a family of 2-groups whose commuting probability converges to 1/2, but no 2-group has commuting probability 1/2 (although $S_3$ does).

But I was wrong! The third conjecture was proved recently in this paper of Thomas Browning: