Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group
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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 .
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(Note: The letters have interchanged roles compared to the formulation of the result on its original page.)
Given: Finite group , groupy -ary relation on . Subgroup of .
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:
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.
- Related MathOverflow question: In an inductive family of groups, does the probability that a particular word is satisfied converge?