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Flag-Transitive 2- Designs Admitting a Two-Dimensional Projective Group

IF 1.3 4区 数学 Q1 MATHEMATICS
Journal of Mathematics Pub Date : 2024-05-03 DOI:10.1155/2024/5521696
Suyun Ding, Yajie Wang, Xiaoqin Zhan
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We have come to the conclusion that if <svg height=\"9.25986pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.01432 11.6478 9.25986\" width=\"11.6478pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> is a nontrivial 2-design having block size 5 and <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 9.02496 8.8423\" width=\"9.02496pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> is a two-dimensional projective special linear group which acts flag-transitively on <svg height=\"9.25986pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.01432 11.6478 9.25986\" width=\"11.6478pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-5\"></use></g></svg> with <span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.34882 19.867 11.7782\" width=\"19.867pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,19.867,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.194,0)\"></path></g></svg><span></span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"25.5861838 -8.34882 6.415 11.7782\" width=\"6.415pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,25.636,0)\"></path></g></svg></span> (mod 4), then <svg height=\"9.25986pt\" style=\"vertical-align:-0.2455397pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.01432 11.6478 9.25986\" width=\"11.6478pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g198-5\"></use></g></svg> is a 2-(11, 5, 2) design, a 2-(11, 5, 12) design, a 2-<span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 21.418 12.7178\" width=\"21.418pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-114\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.787,0)\"></path></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"24.2731838 -9.28833 9.204 12.7178\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,24.323,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.563,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"35.6561838 -9.28833 9.205 12.7178\" width=\"9.205pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,35.706,0)\"><use xlink:href=\"#g113-54\"></use></g><g transform=\"matrix(.013,0,0,-0.013,41.947,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"47.0401838 -9.28833 46.186 12.7178\" width=\"46.186pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,47.09,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,53.33,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,57.828,0)\"><use xlink:href=\"#g113-114\"></use></g><g transform=\"matrix(.013,0,0,-0.013,67.116,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,77.653,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,83.893,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,88.391,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> design with <span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.34882 17.646 11.7782\" width=\"17.646pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-114\"></use></g><g transform=\"matrix(.013,0,0,-0.013,10.015,0)\"><use xlink:href=\"#g117-35\"></use></g></svg><span></span><svg height=\"11.7782pt\" style=\"vertical-align:-3.42938pt\" version=\"1.1\" viewbox=\"21.2281838 -8.34882 6.415 11.7782\" width=\"6.415pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.278,0)\"></path></g></svg></span> (mod 4) or a 2-<span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 21.418 12.7178\" width=\"21.418pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.498,0)\"><use xlink:href=\"#g113-114\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.787,0)\"><use xlink:href=\"#g117-36\"></use></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"24.2731838 -9.28833 9.204 12.7178\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,24.323,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,30.563,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"35.6561838 -9.28833 9.205 12.7178\" width=\"9.205pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,35.706,0)\"><use xlink:href=\"#g113-54\"></use></g><g transform=\"matrix(.013,0,0,-0.013,41.947,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"47.0401838 -9.28833 40.417 12.7178\" width=\"40.417pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,47.09,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,51.588,0)\"><use xlink:href=\"#g113-114\"></use></g><g transform=\"matrix(.013,0,0,-0.013,60.876,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,71.413,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,77.653,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,82.151,0)\"></path></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"87.4621838 -9.28833 11.14 12.7178\" width=\"11.14pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,87.512,0)\"><use xlink:href=\"#g113-52\"></use></g><g transform=\"matrix(.013,0,0,-0.013,93.752,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> design with <span><svg height=\"15.6876pt\" style=\"vertical-align:-3.4294pt\" version=\"1.1\" viewbox=\"-0.0498162 -12.2582 17.646 15.6876\" width=\"17.646pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-114\"></use></g><g transform=\"matrix(.013,0,0,-0.013,10.015,0)\"></path></g></svg><span></span><svg height=\"15.6876pt\" style=\"vertical-align:-3.4294pt\" version=\"1.1\" viewbox=\"21.2281838 -12.2582 12.796 15.6876\" width=\"12.796pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.278,0)\"><use xlink:href=\"#g113-51\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,27.518,-5.741)\"></path></g></svg></span> (where <span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 19.615 12.7178\" width=\"19.615pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,11.984,0)\"></path></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"23.1971838 -9.28833 6.417 12.7178\" width=\"6.417pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,23.247,0)\"><use xlink:href=\"#g113-51\"></use></g></svg></span> is an even).","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/5521696","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

The focus of this study is to classify flag-transitive 2-designs. We have come to the conclusion that if is a nontrivial 2-design having block size 5 and is a two-dimensional projective special linear group which acts flag-transitively on with (mod 4), then is a 2-(11, 5, 2) design, a 2-(11, 5, 12) design, a 2- design with (mod 4) or a 2- design with (where is an even).
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可容纳二维投影群的旗反式 2- 设计
本研究的重点是对旗反 2 设计进行分类。我们得出的结论是：如果是一个具有 5 块大小的非小二设计，并且是一个二维投影特殊线性群，且该群旗反式地作用于与（模 4），那么与就是一个 2-（11，5，2）设计、一个 2-（11，5，12）设计、一个与（模 4）的 2-设计或一个与（其中与是偶数）的 2-设计。
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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来源期刊
Journal of Mathematics
Journal of Mathematics Mathematics-General Mathematics
CiteScore
2.50
自引率
14.30%
发文量
0
期刊介绍： Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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