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 ${\displaystyle G}$ 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 ${\displaystyle G}$.

Results

• 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 ${\displaystyle \omega ^{\omega }}$.

Generalizations

• The commuting probability can be defined for other algebraic structures such as finite rings. The 5/8 theorem also applies to finite rings.
• The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.