Commuting probability

In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) 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 $$G$$ be a finite group. We define $$p(G)$$ as the averaged number of pairs of elements of $$G$$ which commute:
 * $$p(G) := \frac{1}{\# G^2} \#\!\left\{ (x,y) \in G^2 \mid xy=yx \right\}$$

where $$\# X$$ denotes the cardinality of a finite set $$X$$.

If one considers the uniform distribution on $$G^2$$, $$p(G)$$ is the probability that two randomly chosen elements of $$G$$ commute. That is why $$p(G)$$ is called the commuting probability of $$G$$.

Results

 * The finite group $$G$$ is abelian if and only if $$p(G) = 1$$.
 * One has
 * $$p(G) = \frac{k(G)}{\# G}$$
 * where $$k(G)$$ is the number of conjugacy classes of $$G$$.


 * If $$G$$ is not abelian then $$p(G) \leq 5/8$$ (this result is sometimes called the 5/8 theorem ) and this upper bound is sharp: there are infinitely many finite groups $$G$$ such that $$p(G) = 5/8$$, the smallest one being the dihedral group of order 8.
 * There is no uniform lower bound on $$p(G)$$. In fact, for every positive integer $$n$$ there exists a finite group $$G$$ such that $$p(G) = 1/n$$.
 * If $$G$$ is not abelian but simple, then $$p(G) \leq 1/12$$ (this upper bound is attained by $$\mathfrak{A}_5$$, the alternating group of degree 5).
 * The set of commuting probabilities of finite groups is reverse-well-ordered, and the reverse of its order type is known to be either $$\omega^\omega$$ or $$\omega^{\omega^2}$$.

Generalizations

 * The commuting probability can be defined for other algebraic structures such as finite rings.
 * The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.