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{1}{# G^{2}} # {(x, y) \in G^{2} ∣ x y = y x}

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{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 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 $𝔄_{5}$ , 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 $ω^{ω}$ .

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.

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.
↑ Das, A. K.; Nath, R. K.; Pournaki, M. R. (2013). "A survey on the estimation of commutativity in finite groups" (PDF). Southeast Asian Bulletin of Mathematics. 37 (2): 161–180.
1 2 Hofmann, Karl H.; Russo, Francesco G. (2012). "The probability that x and y commute in a compact group". Mathematical Proceedings of the Cambridge Philosophical Society. 153 (3): 557–571. arXiv:1001.4856. Bibcode:2012MPCPS.153..557H. doi:10.1017/S0305004112000308. S2CID 115180549.
1 2 Machale, Desmond (1976). "Commutativity in Finite Rings". The American Mathematical Monthly. 83: 30–32. doi:10.1080/00029890.1976.11994032.
↑ Baez, John C. (2018-09-16). "The 5/8 Theorem". Azimut.
↑ Eberhard, Sean (2015). "Commuting probabilities of finite groups". Bulletin of the London Mathematical Society. 47 (5): 796–808. arXiv:1411.0848. doi:10.1112/blms/bdv050. S2CID 119636430.
↑ Browning, Thomas (2023). "Limit points of commuting probabilities of finite groups". Bulletin of the London Mathematical Society. 55 (3): 1392–1403. arXiv:2201.09402. doi:10.1112/blms.12799.
↑ Dutta, Jutirekha; Basnet, Dhiren; Nath, Rajat (2017). "On commuting probability of finite rings". Indagationes Mathematicae. 28: 372–382. arXiv:1510.08211. doi:10.1016/j.indag.2016.10.002.

[1] 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.

[2] Das, A. K.; Nath, R. K.; Pournaki, M. R. (2013). "A survey on the estimation of commutativity in finite groups" (PDF). Southeast Asian Bulletin of Mathematics. 37 (2): 161–180.

[:0-3] 1 2 Hofmann, Karl H.; Russo, Francesco G. (2012). "The probability that x and y commute in a compact group". Mathematical Proceedings of the Cambridge Philosophical Society. 153 (3): 557–571. arXiv:1001.4856. Bibcode:2012MPCPS.153..557H. doi:10.1017/S0305004112000308. S2CID 115180549.

[:1-4] 1 2 Machale, Desmond (1976). "Commutativity in Finite Rings". The American Mathematical Monthly. 83: 30–32. doi:10.1080/00029890.1976.11994032.

[5] Baez, John C. (2018-09-16). "The 5/8 Theorem". Azimut.

[6] Eberhard, Sean (2015). "Commuting probabilities of finite groups". Bulletin of the London Mathematical Society. 47 (5): 796–808. arXiv:1411.0848. doi:10.1112/blms/bdv050. S2CID 119636430.

[7] Browning, Thomas (2023). "Limit points of commuting probabilities of finite groups". Bulletin of the London Mathematical Society. 55 (3): 1392–1403. arXiv:2201.09402. doi:10.1112/blms.12799.

[8] Dutta, Jutirekha; Basnet, Dhiren; Nath, Rajat (2017). "On commuting probability of finite rings". Indagationes Mathematicae. 28: 372–382. arXiv:1510.08211. doi:10.1016/j.indag.2016.10.002.