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.
Let be a finite group. We define as the averaged number of pairs of elements of which commute:
If one consider the uniform distribution on , is the probability that two randomly chosen elements of commute. That is why is called the commuting probability of .
- The finite group is abelian if and only if .
- One has
- where is the number of conjugacy classes of .
- If is not abelian, then (this result is sometimes called the 5/8 theorem) and this upper bound is sharp: there is an infinity of finite groups such that , the smallest one is the dihedral group of order 8.
- There is no uniform lower bound on . In fact, for every positive integer , there exists a finite group such that .
- If is not abelian but simple, then (this upper bound is attained by , the alternating group of degree 5).
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