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
Let [math]\displaystyle{ G }[/math] be a finite group. We define [math]\displaystyle{ p(G) }[/math] as the averaged number of pairs of elements of [math]\displaystyle{ G }[/math] which commute:
- [math]\displaystyle{ p(G) := \frac{1}{\# G^2} \#\!\left\{ (x,y) \in G^2 \mid xy=yx \right\} }[/math]
where [math]\displaystyle{ \# X }[/math] denotes the cardinality of a finite set [math]\displaystyle{ X }[/math].
If one considers the uniform distribution on [math]\displaystyle{ G^2 }[/math], [math]\displaystyle{ p(G) }[/math] is the probability that two randomly chosen elements of [math]\displaystyle{ G }[/math] commute. That is why [math]\displaystyle{ p(G) }[/math] is called the commuting probability of [math]\displaystyle{ G }[/math].
Results
- The finite group [math]\displaystyle{ G }[/math] is abelian if and only if [math]\displaystyle{ p(G) = 1 }[/math].
- One has
- [math]\displaystyle{ p(G) = \frac{k(G)}{\# G} }[/math]
- where [math]\displaystyle{ k(G) }[/math] is the number of conjugacy classes of [math]\displaystyle{ G }[/math].
- If [math]\displaystyle{ G }[/math] is not abelian then [math]\displaystyle{ p(G) \leq 5/8 }[/math] (this result is sometimes called the 5/8 theorem[5]) and this upper bound is sharp: there are infinitely many finite groups [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ p(G) = 5/8 }[/math], the smallest one being the dihedral group of order 8.
- There is no uniform lower bound on [math]\displaystyle{ p(G) }[/math]. In fact, for every positive integer [math]\displaystyle{ n }[/math] there exists a finite group [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ p(G) = 1/n }[/math].
- If [math]\displaystyle{ G }[/math] is not abelian but simple, then [math]\displaystyle{ p(G) \leq 1/12 }[/math] (this upper bound is attained by [math]\displaystyle{ \mathfrak{A}_5 }[/math], 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 [math]\displaystyle{ \omega^\omega }[/math] or [math]\displaystyle{ \omega^{\omega^2} }[/math].[6]
Generalizations
- The commuting probability can be defined for other algebraic structures such as finite rings.[4]
- The commuting probability can be defined for infinite compact groups; the probability measure is then, after a renormalisation, the Haar measure.[3]
References
- ↑ 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.
- ↑ 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.
- ↑ 3.0 3.1 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. doi:10.1017/S0305004112000308. Bibcode: 2012MPCPS.153..557H.
- ↑ 4.0 4.1 Machale, Desmond (1976). "Commutativity in Finite Rings". The American Mathematical Monthly 83: 30–32. doi:10.1080/00029890.1976.11994032.
- ↑ Baez, John C. (2018-09-16). "The 5/8 Theorem". https://johncarlosbaez.wordpress.com/2018/09/16/the-5-8-theorem/.
- ↑ Eberhard, Sean (2015). "Commuting probabilities of finite groups". Bulletin of the London Mathematical Society 47 (5): 796–808. doi:10.1112/blms/bdv050.
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