# 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:

$p(G):={\frac {1}{\#G^{2}}}\#\left\{(x,y)\in G^{2}\colon xy=yx\right\}.$

If one consider 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 is an infinity of finite groups $G$  such that $p(G)=5/8$ , the smallest one is 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).