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{"title":"零势ℚ[𝑥]幂群的可分性特性 II","authors":"Stephen Majewicz, Marcos Zyman","doi":"10.1515/jgth-2023-0288","DOIUrl":null,"url":null,"abstract":"In this paper, we study nilpotent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-powered groups that satisfy the following property: for some set of primes 𝜔 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, every <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ω</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0003.png\"/> <jats:tex-math>\\omega^{\\prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-isolated <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroup in some family of its <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroups is finite 𝜔-type separable.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"8 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Separability properties of nilpotent ℚ[𝑥]-powered groups II\",\"authors\":\"Stephen Majewicz, Marcos Zyman\",\"doi\":\"10.1515/jgth-2023-0288\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we study nilpotent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0288_ineq_0001.png\\\"/> <jats:tex-math>\\\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-powered groups that satisfy the following property: for some set of primes 𝜔 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0288_ineq_0001.png\\\"/> <jats:tex-math>\\\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, every <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>ω</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0288_ineq_0003.png\\\"/> <jats:tex-math>\\\\omega^{\\\\prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-isolated <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0288_ineq_0001.png\\\"/> <jats:tex-math>\\\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroup in some family of its <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">Q</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_jgth-2023-0288_ineq_0001.png\\\"/> <jats:tex-math>\\\\mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroups is finite 𝜔-type separable.\",\"PeriodicalId\":50188,\"journal\":{\"name\":\"Journal of Group Theory\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-08-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Group Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2023-0288\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Group Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0288","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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