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{"title":"Flag-Transitive 2- Designs Admitting a Two-Dimensional Projective Group","authors":"Suyun Ding, Yajie Wang, Xiaoqin Zhan","doi":"10.1155/2024/5521696","DOIUrl":null,"url":null,"abstract":"The focus of this study is to classify flag-transitive 2-designs. 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}
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