The probability that two uniform random elements of a finite group commute with each other
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 formula_1 be a finite group. We define formula_2 as the averaged number of pairs of elements of formula_1 which commute:
formula_4
where formula_5 denotes the cardinality of a finite set formula_6.
If one considers the uniform distribution on formula_7, formula_2 is the probability that two randomly chosen elements of formula_1 commute. That is why formula_2 is called the commuting probability of formula_1.
formula_14
 where formula_15 is the number of conjugacy classes of formula_1.
References.
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