Commuting probability: Difference between revisions

Probability that two elements of a group commute

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.^[1][2] 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,^[3] and can also be generalized to other algebraic structures such as rings.^[4]

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 ${\displaystyle G}$.