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.[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

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Let be a finite group. We define as the averaged number of pairs of elements of which commute:

where denotes the cardinality of a finite set .

If one considers the uniform distribution on , is the probability that two randomly chosen elements of commute. That is why is called the commuting probability of .

Results

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  • The finite group is abelian if and only if .
  • One has
where is the number of conjugacy classes of .
  • If is not abelian then (this result is sometimes called the 5/8 theorem[5]) and this upper bound is sharp: there are infinitely many finite groups such that , the smallest one being the dihedral group of order 8.
  • There is no uniform lower bound on . In fact, for every positive integer there exists a finite group such that .
  • If is not abelian but simple, then (this upper bound is attained by , 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 or .[6]

Generalizations

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References

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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.
  3. ^ a b 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. ^ a b 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.