Commuting probability

In mathematics, the commuting probability of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to be abelian a finite group is.

This concept can be generalized to others algebraic structures such as finite rings.

Definition

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

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

If one consider 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 groupe $G$ is abelian if and only if $p(G)=1$.
• One has

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

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 is an infinity of finite groups $G$ such that $p(G)=5/8$, the smallest one is 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 abelian but simple, then $p(G)\leq 1/12$ (this upper bound is attained by ${\mathfrak {A}}_{5}$, the alternating group of degree 5).

References

• Gustafson, W. H. (1973). "What is the Probability that Two Group Elements Commute?". The American Mathematical Monthly. 80 (9): 1031–1034. doi:10.1080/00029890.1973.11993437.
• Machale, Desmond (1974). "How Commutative Can a Non-Commutative Group Be?". The Mathematical Gazette. 58 (405): 199–202. doi:10.2307/3615961. JSTOR 3615961.
• Guralnick, Robert M.; Robinson, Geoffrey R. (2006). "On the commuting probability in finite groups". Journal of Algebra. 300 (2): 509–528. doi:10.1016/j.jalgebra.2005.09.044.
• Baez, John C. (2018-09-16). "The 5/8 Theorem". Azimut.