Muhammad Ashiq , Tahir Imran , Muhammad Asad Zaighum
{"title":"Δ(3,n,k)在投影线上的作用","authors":"Muhammad Ashiq , Tahir Imran , Muhammad Asad Zaighum","doi":"10.1016/j.trmi.2017.09.005","DOIUrl":null,"url":null,"abstract":"<div><p>Each conjugacy class of actions of the triangle group <span><math><mi>Δ</mi><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> over the projective line <span><math><mi>P</mi><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span> can be represented by a coset diagram <span><math><mi>D</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>θ</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and <span><math><mi>q</mi></math></span> is a prime number. In this paper, we have considered conjugacy classes which arise from the actions of <span><math><mi>Δ</mi><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>〈</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mi>r</mi><mi>s</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mn>1</mn><mo>〉</mo></mrow></math></span> over <span><math><mi>P</mi><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is finite field. The points of <span><math><mi>P</mi><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span> are the elements of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> together with the additional point <span><math><mi>∞</mi><mo>.</mo></math></span>\n</p></div>","PeriodicalId":43623,"journal":{"name":"Transactions of A Razmadze Mathematical Institute","volume":"172 1","pages":"Pages 1-6"},"PeriodicalIF":0.3000,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.trmi.2017.09.005","citationCount":"1","resultStr":"{\"title\":\"Actions of Δ(3,n,k) on projective line\",\"authors\":\"Muhammad Ashiq , Tahir Imran , Muhammad Asad Zaighum\",\"doi\":\"10.1016/j.trmi.2017.09.005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Each conjugacy class of actions of the triangle group <span><math><mi>Δ</mi><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> over the projective line <span><math><mi>P</mi><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span> can be represented by a coset diagram <span><math><mi>D</mi><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>θ</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and <span><math><mi>q</mi></math></span> is a prime number. In this paper, we have considered conjugacy classes which arise from the actions of <span><math><mi>Δ</mi><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>〈</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mi>s</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mi>r</mi><mi>s</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mn>1</mn><mo>〉</mo></mrow></math></span> over <span><math><mi>P</mi><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is finite field. The points of <span><math><mi>P</mi><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span> are the elements of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> together with the additional point <span><math><mi>∞</mi><mo>.</mo></math></span>\\n</p></div>\",\"PeriodicalId\":43623,\"journal\":{\"name\":\"Transactions of A Razmadze Mathematical Institute\",\"volume\":\"172 1\",\"pages\":\"Pages 1-6\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2018-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.trmi.2017.09.005\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of A Razmadze Mathematical Institute\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2346809217300715\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of A Razmadze Mathematical Institute","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2346809217300715","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Each conjugacy class of actions of the triangle group over the projective line can be represented by a coset diagram , where and is a prime number. In this paper, we have considered conjugacy classes which arise from the actions of over , where is finite field. The points of are the elements of together with the additional point
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