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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]

DefinitionEdit

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

 

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

ResultsEdit

  • 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 is an infinity of finite groups   such that  , the smallest one is 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).

GeneralizationsEdit

ReferencesEdit

  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. doi:10.1017/S0305004112000308.
  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.