Probability of commuting

A couple years ago I wrote a blog post looking at how close the quaternions come to commuting. That is, the post looked at the average norm of xy - yx.

A related question would be to ask how often quaternions do commute, i.e. the probability that xy - yx = 0 for randomly chosen x and y.

There's a general theorem for this [1]. For a discrete non-abelian group, the probability that two elements commute, chosen uniformly at random, is never more than 5/8 for any group.

To put it another way, in a finite group either all pairs of elements commute with each other or no more than 5/8 of all pairs commute, with no possibilities in between. You can't have a group, for example, in which exactly 3 out of 4 pairs commute.

What if we have an infinite group like the quaternions?

Before we can answer that, we've got to say how we'd compute probabilities. With a finite group, the natural thing to do is make every point have equal probability. For a (locally compact) infinite group the natural choice is Haar measure.

Subject to some technical conditions, Haar measure is the only measure that interacts as expected with the group structure. It's unique up to a constant multiple, and so it's unique when we specify that the measure of the whole group has to be 1.

For compact non-abelian groups with Haar measure, we again get the result that no more than 5/8 of pairs commute.

[1] W. H. Gustafson. What is the Probability that Two Group Elements Commute? The American Mathematical Monthly, Nov., 1973, Vol. 80, No. 9, pp. 1031-1034.

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