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 \#\!\left\$
where $\# X$ denotes the cardinality of a finite set $X$.

If one considers 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
::$p(G) = \frac$
: 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 are infinitely many finite groups $G$ such that $p(G) = 5/8$, the smallest one being 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 $\mathfrak_5$, 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 $\omega^\omega$ or $\omega^$.

Generalizations

* 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.

References

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Finite groups