# 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.

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## Contents

## Statement

### In fraction terms

For a group , define to be the set:

Then, if is a subgroup of a finite group , we have:

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

For a finite group , let denote the Number of conjugacy classes (?) in . Then, if is a subgroup of , we have:

Equivalently:

## Related facts

- 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

## Facts used

- 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)

## Proof

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.