- 1 Definition
- 2 Facts
- 3 Particular cases
- 4 References
In terms of probability of satisfaction of a word
The commuting fraction or commuting probability or commutativity ratio of a finite group is defined as the probability of satisfaction of the commutator as a word. Note that it does not matter which of the conventions (left or right) we choose to define the commutator. With the left convention, the commutator is , and with the right convention, the commutator is .
In terms of fraction of pairs that commute
For a finite group , define:
The commuting fraction or commuting probability or commutativity ratio of is the quotient:
It can also be viewed as the probability that two elements of the group, picked independently uniformly at random, commute.
So the commuting fraction of can be written as:
Note that this can also be written as follows, where is the center of :
This simplifies to:
In terms of number of conjugacy classes
In terms of index of centralizers
This formulation has the advantage that, in addition to making sense for a finite group, it also makes sense for a FZ-group, i.e., a group whose center has finite index, i.e., the inner automorphism group is a finite group. If the whole group is finite, it agrees with the usual definition.
The commuting fraction is defined as:
Also, since any two elements in the same coset of in have the same centralizer, depends only on the coset of , and the sum is hence independent of the choice of .
Equivalence of definitions
Further information: Equivalence of definitions of commuting fraction
The equivalence follows from the orbit-counting theorem.
Relation with subgroups, quotients, and direct products
- Commuting fraction in subgroup is at least as much as in whole group
- Number of conjugacy classes in a subgroup of finite index is bounded by index times number of conjugacy classes in the whole group
- Commuting fraction in quotient group is at least as much as in whole group
- Commuting fraction of direct product is product of commuting fractions
- Commuting fraction of extension group is bounded by product of commuting fractions of normal subgroup and quotient group
High values of commuting fraction and indication of abelianness
- A group is abelian if and only if its commuting fraction is .
- Commuting fraction more than five-eighths implies abelian
- Commuting fraction equals five-eighths iff inner automorphism group is Klein four-group
Here are the commuting fractions for some non-abelian groups of small order. Note that all abelian groups have commuting fraction equal to , so these are omitted from the list. Note that the commuting fraction is the quotient of the number of conjugacy classes to the order of the group.