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We demonstrate that the size of the quotient set in a finite group exceeds 1.67 times the size of the original subset, suggesting new insights into cosets.
We show that for a finite, nonempty subset A of a group, the quotient set A −1 A ≔ { a 1 −1 a 2 : a 1 , a 2 ∈ A } has size A⁻¹ A ≥ 5 3 A ∣ A − 1 A ∣≥ 5 3 ∣ A ∣ , unless A is densely contained in a coset, or in a union of two cosets of a finite subgroup.
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Vsevolod F. Lev (2025) studied this question.
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