Commuting probability

Short description: The probability that two uniform random elements of a finite group commute with each other

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:

$p(G):=\frac{1}{\mathrm{\#}{G}^2}\mathrm{\#}\{(x,y)\in G^2\mid xy=yx\}$

where $\mathrm{\#}X$ 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

$p(G)=\frac{k(G)}{\mathrm{\#}G}$

where $k(G)$ is the number of conjugacy classes of $G$.

• If $G$ is not abelian then $p(G)\le 5/8$ (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 $p(G)\le 1/12$ (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.