# Fraction of tuples satisfying groupy relation 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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## Statement

Suppose is a finite group. Suppose is a -ary relation on with the property that if we consider a -tuple , and we fix all coordinates except the coordinate, the set of possibilities for forms a subgroup of .

Suppose is a subgroup of . Then, we have:

In other words, the fraction of -tuples from satisfying is at least as much as the fraction of -tuples in satisfying .

## Related facts

### Applications

- Commuting fraction in subgroup is at least as much as in whole group
- Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
- Commuting fraction in subring of finite Lie ring is at least as much as in whole ring
- Associating fraction in subring of finite non-associative ring is at least as much as in whole ring
- Fraction of tuples for iterated Lie bracket word in subring of finite Lie ring is at least as much as in whole ring

## Facts used

- Index satisfies transfer inequality: This states that if , then . However, the form in which we use it is the conditional probability formulation, which states that:

(Note: The letters have interchanged roles compared to the formulation of the result on its original page.)

## Proof

**Given**: Finite group , groupy -ary relation on . Subgroup of .

**To prove**:

**Proof**: We prove the statement by induction on . The proof for the base case and the induction step are similar, but we give both proofs.

**Proof of base case**: In the base case, the *relation* has only one element, which means that it is just a subset of . The groupiness now says that this subset is a subgroup.Then, fact (1) yields that:

Inverting both sides and flipping the inequality sign yields:

**Proof of induction step**: Suppose we have shown the result for all -ary relations. We now need to show it for the -ary relation .

Define as the subset of comprising those values of such that . Each such is a subgroup, and:

Also, we have:

Dividing both sides of (1) by and both sides of (2) by , we get respectively:

By fact (1), we have , yielding:

Thus, each of the summands to (4) is bigger than the corresponding summand to (3), and thus we get:

Multiplying both denominators by , we get:

For , define as the relation on induced by with the last coordinate . Then:

Finally, we have:

Dividing (6) and (7) by and respectively, we get:

and:

Each is groupy. Thus, by induction, each satisfies:

Plugging (10) into the summations on the right sides of (8) and (9), we get:

Combining (5) and (11) yields the desired results.

## External links

- Related MathOverflow question: In an inductive family of groups, does the probability that a particular word is satisfied converge?