{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","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-2023-0288","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
In this paper, we study nilpotent Q[x]\mathbb{Q}[x]-powered groups that satisfy the following property: for some set of primes 𝜔 in Q[x]\mathbb{Q}[x], every ω′\omega^{\prime}-isolated Q[x]\mathbb{Q}[x]-subgroup in some family of its Q[x]\mathbb{Q}[x]-subgroups is finite 𝜔-type separable.
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