# Index satisfies transfer inequality

## Contents

## Statement

### In terms of index

Suppose is a group and are subgroups of . Then:

.

### In terms of conditional probability

This formulation is valid for finite groups. It says that if is a group and are subgroups, then:

In other words, what it says is that, for a uniform distribution on a finite group, knowing that a particular element is in the subgroup either increases or keeps the same the probability that the element is in the subgroup .

## Related facts

### Applications

The formulation in terms of conditional probability is particularly useful to prove results on the fractions of tuples satisfying a groupy relation. See, for instance:

- Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group
- Fraction of ordered pairs commuting in subgroup is at least as much as in whole group

## Facts used

- Product formula: if are subgroups, there is a natural bijection between the left cosets of in and the left cosets of in .

## Proof

**Given**: A group with subgroups .

**To prove**: .

**Proof**: By fact (1), the number of left cosets of in equals the number of left cosets of in . Thus, the number of left cosets of in is *at least* as much as the number of left cosets of in , yielding the desired inequality.