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

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

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

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

Results

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

${\displaystyle p(G)={\frac {k(G)}{\#G}}}$

where ${\displaystyle k(G)}$ is the number of conjugacy classes of ${\displaystyle G}$.

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

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" (PDF). Southeast Asian Bulletin of Mathematics. 37 (2): 161–180.
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.
4. 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.