Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2024_434_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation> be even, and let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2024_434_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="286" /> </InlineMediaObject> <EquationSource Format="TEX">$G = \langle x, y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∣</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>y</mi> <mrow> <mi>n</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>y</mi> <mi>n</mi> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>y</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <msup> <mi>y</mi> <mi>s</mi> </msup> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2024_434_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$s^2 \equiv 1 \pmod n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2024_434_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$s \not \equiv \pm 1 \pmod n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≢</mo> <mo>±</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we provide the precise values of some zero-sum constants over G, namely the small Davenport constant, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2024_434_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-constant, Gao constant, and Erdős-Ginzburg-Ziv constant. In particular, the Gao’s and Zhuang-Gao’s Conjectures hold for G. We also solve the associated inverse problems when <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2024_434_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$n \equiv 0 \pmod 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Some zero-sum problems over $\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle $

期刊论文
发表日期：

2025年1月10日

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S. Ribas

摘要

Let $n \ge 8$ $n \geq 8$ be even, and let $G = \langle x, y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle $ $G = ⟨ x, y ∣ x^{2} = y^{n / 2}, y^{n} = 1, y x = x y^{s} ⟩$ , where $s^2 \equiv 1 \pmod n$ $s^{2} \equiv 1 (mod n)$ and $s \not \equiv \pm 1 \pmod n$ $s ≢ \pm 1 (mod n)$ . In this paper, we provide the precise values of some zero-sum constants over G, namely the small Davenport constant, $\eta $ $η$ -constant, Gao constant, and Erdős-Ginzburg-Ziv constant. In particular, the Gao’s and Zhuang-Gao’s Conjectures hold for G. We also solve the associated inverse problems when $n \equiv 0 \pmod 4$ $n \equiv 0 (mod 4)$ .

Some zero-sum problems over \(\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle \)

摘要