{"title":"关于非分裂元环群上的直接和逆零和问题","authors":"","doi":"10.1016/j.disc.2024.114213","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>G</mi><mo>=</mo><mrow><mo>〈</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>x</mi><mi>y</mi><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>ℓ</mi></mrow></msup><mo>〉</mo></mrow></math></span> be the non-split metacyclic group with <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>2</mn><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>ℓ</mi><mo>≢</mo><mo>±</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>2</mn><mi>n</mi><mo>)</mo></math></span>. In this paper, we obtain the exact values of small Davenport constant <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, Gao constant <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <em>η</em>-constant <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and Erdős-Ginzburg-Ziv constant <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Additionally, we study the associated inverse problems on <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In 2003, Gao conjectured that <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mtext>exp</mtext><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any finite group <em>G</em>. In 2005, Gao and Zhuang conjectured that <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo></math></span> for any finite group <em>G</em>. As a result, we confirm the two conjectures for non-split metacyclic groups.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the direct and inverse zero-sum problems over non-split metacyclic group\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114213\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span><math><mi>G</mi><mo>=</mo><mrow><mo>〈</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>x</mi><mi>y</mi><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>ℓ</mi></mrow></msup><mo>〉</mo></mrow></math></span> be the non-split metacyclic group with <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>2</mn><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>ℓ</mi><mo>≢</mo><mo>±</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mspace></mspace><mo>(</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><mn>2</mn><mi>n</mi><mo>)</mo></math></span>. In this paper, we obtain the exact values of small Davenport constant <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, Gao constant <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <em>η</em>-constant <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and Erdős-Ginzburg-Ziv constant <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Additionally, we study the associated inverse problems on <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In 2003, Gao conjectured that <span><math><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mtext>exp</mtext><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any finite group <em>G</em>. In 2005, Gao and Zhuang conjectured that <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo></math></span> for any finite group <em>G</em>. As a result, we confirm the two conjectures for non-split metacyclic groups.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-08-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24003443\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003443","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the direct and inverse zero-sum problems over non-split metacyclic group
Let be the non-split metacyclic group with and . In this paper, we obtain the exact values of small Davenport constant , Gao constant , η-constant and Erdős-Ginzburg-Ziv constant . Additionally, we study the associated inverse problems on , , and . In 2003, Gao conjectured that for any finite group G. In 2005, Gao and Zhuang conjectured that for any finite group G. As a result, we confirm the two conjectures for non-split metacyclic groups.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.