Commuting probability

In mathematics and more precisely in group theory, the commuting probability of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite groups equipped with a suitable probability measure, and can also be generalized to other algebraic structures such as rings.

Definition

Let be a finite group. We define as the averaged number of pairs of elements of which commute:
where denotes the cardinality of a finite set.
If one considers the uniform distribution on, is the probability that two randomly chosen elements of commute. That is why is called the commuting probability of.

Results

The finite group is abelian if and only if.
One has
If is not abelian then and this upper bound is sharp: there are infinitely many finite groups such that, the smallest one being the dihedral group of order 8.
There is no uniform lower bound on. In fact, for every positive integer there exists a finite group such that.
If is not abelian but simple, then .
The set of commuting probabilities of finite groups is reverse-well-ordered, and the reverse of its order type is.

Generalizations

The commuting probability can be defined for other algebraic structures such as finite rings. The 5/8 theorem also applies to finite rings.
The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.