Anakin Dey, Kolton O’Neal, Duc Van Khanh Tran, Camron Upshur, Yong Yang
{"title":"秩为 5 和 6 的原始可解置换群的分类","authors":"Anakin Dey, Kolton O’Neal, Duc Van Khanh Tran, Camron Upshur, Yong Yang","doi":"10.1515/jgth-2023-0205","DOIUrl":null,"url":null,"abstract":"Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0205_ineq_0001.png\"/> <jats:tex-math>G_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0205_ineq_0001.png\"/> <jats:tex-math>G_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in Ω, including the trivial orbit <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0205_ineq_0003.png\"/> <jats:tex-math>\\{\\alpha\\}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Classifying primitive solvable permutation groups of rank 5 and 6\",\"authors\":\"Anakin Dey, Kolton O’Neal, Duc Van Khanh Tran, Camron Upshur, Yong Yang\",\"doi\":\"10.1515/jgth-2023-0205\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0205_ineq_0001.png\\\"/> <jats:tex-math>G_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0205_ineq_0001.png\\\"/> <jats:tex-math>G_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in Ω, including the trivial orbit <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo stretchy=\\\"false\\\">{</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\\\"false\\\">}</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0205_ineq_0003.png\\\"/> <jats:tex-math>\\\\{\\\\alpha\\\\}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2023-0205\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0205","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Classifying primitive solvable permutation groups of rank 5 and 6
Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G0G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G0G_{0} in Ω, including the trivial orbit {α}\{\alpha\}. In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.
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