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