{"title":"A note on block-transitive 2 − (v,k,1) designs","authors":"Shangzhao Li, Xiran Zhang, Jing Chen, Shaojun Dai","doi":"10.1016/j.amc.2025.129314","DOIUrl":null,"url":null,"abstract":"Let <ce:italic>G</ce:italic> act block-transitively on a <mml:math altimg=\"si1.svg\"><mml:mn>2</mml:mn><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mn>12</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> design. If <ce:italic>G</ce:italic> is solvable, then (1) if <mml:math altimg=\"si2.svg\"><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>v</mml:mi></mml:math>, then <mml:math altimg=\"si27.svg\"><mml:mi>v</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:msup><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>, <mml:math altimg=\"si28.svg\"><mml:mi>G</mml:mi><mml:mo>≤</mml:mo><mml:mi>A</mml:mi><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math altimg=\"si29.svg\"><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"0.40em\"></mml:mspace><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">mod</mml:mi></mml:mrow><mml:mspace width=\"0.25em\"></mml:mspace><mml:mn>44</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>; (2) if <mml:math altimg=\"si30.svg\"><mml:mn>3</mml:mn><mml:mo>∤</mml:mo><mml:mi>v</mml:mi></mml:math>, then <mml:math altimg=\"si20.svg\"><mml:mi>v</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>, <mml:math altimg=\"si31.svg\"><mml:mi>G</mml:mi><mml:mo>≤</mml:mo><mml:mi>A</mml:mi><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math altimg=\"si32.svg\"><mml:mi>p</mml:mi><mml:mo>≠</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math altimg=\"si10.svg\"><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"0.40em\"></mml:mspace><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">mod</mml:mi></mml:mrow><mml:mspace width=\"0.25em\"></mml:mspace><mml:mn>132</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>.","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"49 1","pages":""},"PeriodicalIF":3.5000,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.amc.2025.129314","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let G act block-transitively on a 2−(v,12,1) design. If G is solvable, then (1) if 3|v, then v=3n, G≤AΓL(1,3n), where pn≡1(mod44); (2) if 3∤v, then v=pn, G≤AΓL(1,pn), where p≠2 and pn≡1(mod132).
期刊介绍：
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.
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