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.

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. 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 . 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 \omega^\omega .

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