• Images
• Videos
• Documents
• Books
• Articles

Definition

Let G {\displaystyle G} be a finite group. We define p ( G ) {\displaystyle p(G)} as the averaged number of pairs of elements of G {\displaystyle G} which commute:

p ( G ) := 1 # G 2 # { ( x , y ) ∈ G 2 ∣ x y = y x } {\displaystyle p(G):={\frac {1}{\#G^{2}}}\#\!\left\{(x,y)\in G^{2}\mid xy=yx\right\}}

where # X {\displaystyle \#X} denotes the cardinality of a finite set X {\displaystyle X} .

If one considers the uniform distribution on G 2 {\displaystyle G^{2}} , p ( G ) {\displaystyle p(G)} is the probability that two randomly chosen elements of G {\displaystyle G} commute. That is why p ( G ) {\displaystyle p(G)} is called the commuting probability of G {\displaystyle G} .

Results

• The finite group G {\displaystyle G} is abelian if and only if p ( G ) = 1 {\displaystyle p(G)=1} .
• One has

p ( G ) = k ( G ) # G {\displaystyle p(G)={\frac {k(G)}{\#G}}} where k ( G ) {\displaystyle k(G)} is the number of conjugacy classes of G {\displaystyle G} .

• If G {\displaystyle G} is not abelian then p ( G ) ≤ 5 / 8 {\displaystyle p(G)\leq 5/8} (this result is sometimes called the 5/8 theorem⁵) and this upper bound is sharp: there are infinitely many finite groups G {\displaystyle G} such that p ( G ) = 5 / 8 {\displaystyle p(G)=5/8} , the smallest one being the dihedral group of order 8.
• There is no uniform lower bound on p ( G ) {\displaystyle p(G)} . In fact, for every positive integer n {\displaystyle n} there exists a finite group G {\displaystyle G} such that p ( G ) = 1 / n {\displaystyle p(G)=1/n} .
• If G {\displaystyle G} is not abelian but simple, then p ( G ) ≤ 1 / 12 {\displaystyle p(G)\leq 1/12} (this upper bound is attained by A 5 {\displaystyle {\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 ω ω {\displaystyle \omega ^{\omega }} or ω ω 2 {\displaystyle \omega ^{\omega ^{2}}} .⁶

Generalizations

• The commuting probability can be defined for other algebraic structures such as finite rings.⁷
• The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.⁸

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. /wiki/Doi_(identifier) ↩

2. Das, A. K.; Nath, R. K.; Pournaki, M. R. (2013). "A survey on the estimation of commutativity in finite groups" (PDF). Southeast Asian Bulletin of Mathematics. 37 (2): 161–180. http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201302&filename=02_37(2).pdf ↩

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. /wiki/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society ↩

4. Machale, Desmond (1976). "Commutativity in Finite Rings". The American Mathematical Monthly. 83: 30–32. doi:10.1080/00029890.1976.11994032. /wiki/Doi_(identifier) ↩

5. Baez, John C. (2018-09-16). "The 5/8 Theorem". Azimut. https://johncarlosbaez.wordpress.com/2018/09/16/the-5-8-theorem/ ↩

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. /wiki/ArXiv_(identifier) ↩

7. Machale, Desmond (1976). "Commutativity in Finite Rings". The American Mathematical Monthly. 83: 30–32. doi:10.1080/00029890.1976.11994032. /wiki/Doi_(identifier) ↩

8. 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. /wiki/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society ↩