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<p>We show that for a finite, nonempty subset <i>A</i> of a group, the quotient set <i>A</i><sup>−1</sup><i>A</i> ≔ {<i>a</i><Stack> <sub>1</sub> <sup>−1</sup> </Stack><i>a</i><sub>2</sub>: <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub> ∈ <i>A</i>} has size <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mid A^{-1} A\mid \geq {5\over 3}\mid A \mid\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">∣</mo> <msup> <mi>A</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <mo>∣≥</mo> <mrow> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>∣</mo> <mi>A</mi> <mo stretchy="false">∣</mo> </math></EquationSource> </InlineEquation>, unless <i>A</i> is densely contained in a coset, or in a union of two cosets of a finite subgroup.</p>

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Quotient sets in nonabelian groups

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  • 发表日期：

    2025年11月30日

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  • Vsevolod F. Lev

摘要

We show that for a finite, nonempty subset A of a group, the quotient set A−1A ≔ {a 1 −1 a2: a1, a2 ∈ A} has size \(\mid A^{-1} A\mid \geq {5\over 3}\mid A \mid\) ∣ 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.