Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

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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 G is a finite group. Suppose R \subseteq G^n is a n-ary relation on G with the property that if we consider a n-tuple (g_1,g_2, \dots, g_i,\dots, g_n), and we fix all coordinates except the i^{th} coordinate, the set of possibilities for g_i forms a subgroup of G.

Suppose H is a subgroup of G. Then, we have:

\! \frac{|H^n \cap R|}{|H^n|} \ge \frac{|R|}{|G^n|}

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

Related facts

Applications

Facts used

  1. Index satisfies transfer inequality: This states that if H,K \le G, then [H:H \cap K] \le [G:K]. However, the form in which we use it is the conditional probability formulation, which states that:

\! \frac{|H \cap K|}{|H|} \ge \frac{|K|}{|G|}

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

Proof

Given: Finite group G, groupy n-ary relation R on G. Subgroup H of G.

To prove: \! \frac{|H^n \cap R|}{|H^n|} \ge \frac{|R|}{|G^n|}

Proof: We prove the statement by induction on n. The proof for the base case n = 1 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 R of G. The groupiness now says that this subset is a subgroup.Then, fact (1) yields that:

\! \frac{|H|}{|H \cap R|} \le \frac{|G|}{|R|}

Inverting both sides and flipping the inequality sign yields:

\! \frac{|H \cap R|}{|H|} \ge \frac{|R|}{|G|}

Proof of induction step: Suppose we have shown the result for all (n-1)-ary relations. We now need to show it for the n-ary relation R.

Define R_{(g_1,g_2,\dots,g_{n-1})} as the subset of G comprising those values of g_n such that (g_1,g_2,\dots,g_{n-1},g_n) \in R. Each such R_{(g_1,g_2,\dots,g_{n-1})} is a subgroup, and:

\! |R| = \sum_{(g_1,g_2,\dots,g_{n-1}) \in G^{n-1}} |R_{(g_1,g_2,\dots,g_{n-1})}|\qquad (1)

Also, we have:

\! |R \cap (G^{n-1} \times H)| = \sum_{(g_1,g_2,\dots,g_{n-1}) \in G^{n-1}} |R_{(g_1,g_2,\dots,g_{n-1})} \cap H| \qquad(2)

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

\! \frac{|R|}{|G|} = \sum_{(g_1,g_2,\dots,g_{n-1}) \in G^{n-1}} \frac{|R_{(g_1,g_2,\dots,g_{n-1})}|}{|G|} \qquad (3)

\! \frac{|R \cap (G^{n-1} \times H)|}{|H|} = \sum_{(g_1,g_2,\dots,g_{n-1}) \in G^{n-1}} \frac{|R_{(g_1,g_2,\dots,g_{n-1})} \cap H|}{|H|} \qquad(4)

By fact (1), we have [H:R_{(g_1,g_2,\dots,g_{n-1})} \cap H] \le [G:R_{(g_1,g_2,\dots,g_{n-1})}], yielding:

\! \frac{|R_{(g_1,g_2,\dots,g_{n-1})} \cap H|}{|H|} \ge \frac{|R_{(g_1,g_2,\dots,g_{n-1})}|}{|G|}

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

\! \frac{|R \cap (G^{n-1} \times H)|}{|H|} \ge \frac{|R|}{|G|}

Multiplying both denominators by |G^{n-1}|, we get:

\! \frac{|R \cap (G^{n-1} \times H)|}{|G^{n-1} \times H|} \ge \frac{|R|}{|G^n|} \qquad (5)

For g \in G, define R_{g,n} as the relation on G^{n-1} induced by R with the last coordinate g. Then:

\! |R \cap (G^{n-1} \times H)| = \sum_{h \in H} |R_{h,n}| \qquad (6)

Finally, we have:

\! |R \cap H^n| = \sum_{h \in H} | R_{h,n} \cap H^{n-1}| \qquad (7)

Dividing (6) and (7) by |G^{n-1} \times H| and |H^n| respectively, we get:

\! \frac{|R \cap (G^{n-1} \times H)|}{|G^{n-1} \times H|} = \frac{1}{|H|}\sum_{h \in H} \frac{|R_{h,n}|}{|G^{n-1}|}\qquad(8)

and:

\! \frac{|R \cap H^n|}{|H^n|} = \frac{1}{|H|} \sum_{h \in H} \frac{|R_{h,n} \cap H^{n-1}|}{|H^{n-1}|}\qquad (9)

Each R_{h,n} is groupy. Thus, by induction, each R_{h,n} satisfies:

\! \frac{|R_{h,n} \cap H^{n-1}|}{|H^{n-1}|}\ge \frac{|R_{h,n}|}{|G^{n-1}|} \qquad (10)

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

\! \frac{|R \cap H^n|}{|H^n|} \ge  \frac{|R \cap (G^{n-1} \times H)|}{|G^{n-1} \times H|} \qquad (11)

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

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