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. 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 {\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 theorem) 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 ω ω {\displaystyle \omega ^{\omega }}.

Generalizations