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.

Let ${\displaystyle G}$be a finite group. We define ${\displaystyle p(G)}$as the averaged number of pairs of elements of ${\displaystyle G}$which commute:

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

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

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

• The finite group ${\displaystyle G}$is abelian if and only if ${\displaystyle p(G)=1}$.
• One has

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

where ${\displaystyle k(G)}$is the number of conjugacy classes of ${\displaystyle G}$.

• If ${\displaystyle G}$is not abelian then ${\displaystyle p(G)\leq 5/8}$(this result is sometimes called the 5/8 theorem) and this upper bound is sharp: there are infinitely many finite groups ${\displaystyle G}$such that ${\displaystyle p(G)=5/8}$, the smallest one being the dihedral group of order 8.
• There is no uniform lower bound on ${\displaystyle p(G)}$. In fact, for every positive integer ${\displaystyle n}$there exists a finite group ${\displaystyle G}$such that ${\displaystyle p(G)=1/n}$.
• If ${\displaystyle G}$is not abelian but simple, then ${\displaystyle p(G)\leq 1/12}$(this upper bound is attained by ${\displaystyle {\mathfrak {A}}_{5}}$, 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 ${\displaystyle \omega ^{\omega }}$or ${\displaystyle \omega ^{\omega ^{2}}}$.

• 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.