Commuting probability

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:

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

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[5]) and this upper bound is sharp: there is an infinity of finite groups ${\displaystyle G}$ such that ${\displaystyle p(G)=5/8}$, the smallest one is 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 commuting probability can be defined for others algebraic structures such as finite rings.[4]
• The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.[3]