Commuting fraction
Definition
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.
Note that:
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
The commuting fraction of a finite group is defined as the quotient of its number of conjugacy classes by its order.
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:
where denotes the index of in , denotes the center of , is a left transversal for in , and is the centralizer of in . Note that this involves a finite sum of finite numbers, hence it makes sense.
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.
Facts
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
Particular cases
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.
References
- Restrictions on commutativity ratios in finite groups by R. Heffernan, D. Machale and A. Ni. She, International Journal of Group Theory, Volume 3,Number 4, Page 1 - 12(Year 2014): ^{Official copy (ungated)}^{More info}