Commuting fraction in subgroup is at least as much as in whole group
This article is about a result whose hypothesis or conclusion has to do with the fraction of group elements or tuples of group elements satisfying a particular condition.
View other such statements
In fraction terms
For a group , define to be the set:
We sometimes use the term Commuting fraction (?) for the quotient for a given finite group . In those terms, the commuting fraction of a subgroup is at least as much as that of the whole group.
In probability terms
The probability that two elements picked uniformly at random commute cannot increase when we pass from a subgroup to the whole group.
In terms of number of conjugacy classes
- Abelianness is subgroup-closed: This states that any subgroup of an abelian group is abelian. Note that for finite abelian groups, the statement is a particular case of the one on this page. Namely, if , and the fraction of ordered pairs commuting in is , then the fraction of ordered pairs commuting in is , hence equal to , hence is also abelian.
- Number of conjugacy classes in a subgroup may be more than in the whole group
- Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group (which in turn uses index satisfies transfer inequality)
The proof follows directly from fact (1) and the observation that the relation of commuting is groupy in both inputs -- for any element, the set of elements commuting with it is a subgroup, called the centralizer of that element.