Joseph’s conjectures about commuting probability

The commuting probability of a finite group is defined to be the probability that two uniformly random elements commute. If has order and conjugacy classes, this is the same thing as . 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 , 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 .

1. every limit point of is rational,
2. for every there is some interval that avoids,
3. every nonzero limit point of is in .

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 does).

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

https://arxiv.org/abs/2201.09402