Commuting probability explained

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, 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) : = (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) = (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^[3]) 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 ak{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}} .^[4]

Generalizations

• The commuting probability can be defined for other algebraic structures such as finite rings.^[5]
• The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.^[6]

Notes and References

1. 10.1080/00029890.1973.11993437. What is the Probability that Two Group Elements Commute?. The American Mathematical Monthly. 80. 9. 1031–1034. 1973. Gustafson. W. H..
2. A survey on the estimation of commutativity in finite groups. Southeast Asian Bulletin of Mathematics. 37. 2. 161–180. 2013. Das. A. K.. Nath. R. K.. Pournaki. M. R..
3. Web site: The 5/8 Theorem. Baez. John C.. 2018-09-16. Azimut.
4. Commuting probabilities of finite groups. Bulletin of the London Mathematical Society. 47. 5. 796–808. 2015. Eberhard. Sean. 10.1112/blms/bdv050 . 1411.0848. 119636430 .
5. 10.1080/00029890.1976.11994032. Commutativity in Finite Rings. The American Mathematical Monthly. 83. 30–32. 1976. Machale. Desmond.
6. 10.1017/S0305004112000308. The probability that x and y commute in a compact group. Mathematical Proceedings of the Cambridge Philosophical Society. 153. 3. 557–571. 2012. Hofmann. Karl H.. Russo. Francesco G.. 1001.4856. 2012MPCPS.153..557H . 115180549 .