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:

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