Commuting probability

Last updated July 28, 2023

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]

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}\mid xy=yx\right\}

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^[5]) 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 ${\mathfrak {A}}_{5}$ , 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 known to be either $\omega ^{\omega }$ or $\omega ^{\omega ^{2}}$ .^[6]

Generalizations

The commuting probability can be defined for other algebraic structures such as finite rings.^[4]
The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.^[3]

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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". 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.

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[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". 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.

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