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

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 \omega^\omega.[6][7]

Generalizations

References

  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.
  3. ^ 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
  4. ^ 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. doi:10.1016/j.indag.2016.10.002