无穷级的适当子群具有可变传递关系的群

Pub Date : 2024-05-17 DOI:10.1515/jgth-2023-0296

Adolfo Ballester Bolinches, Maria De Falco, Francesco de Giovanni, Carmela Musella

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The property \n \n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>PT</m:mi>\n </m:math>\n \n \\mathrm{PT}\n \n is not inherited by subgroups, and 𝐺 is called a \n \n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mover accent=\"true\">\n <m:mi>PT</m:mi>\n <m:mo>̄</m:mo>\n </m:mover>\n </m:math>\n \n \\overline{\\mathrm{PT}}\n \n -group if all its subgroups have the \n \n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>PT</m:mi>\n </m:math>\n \n \\mathrm{PT}\n \n -property.\nWe prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the \n \n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>PT</m:mi>\n </m:math>\n \n \\mathrm{PT}\n \n -property, then 𝐺 is a \n \n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mover accent=\"true\">\n <m:mi>PT</m:mi>\n <m:mo>̄</m:mo>\n </m:mover>\n </m:math>\n \n \\overline{\\mathrm{PT}}\n \n -group.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Groups whose proper subgroups of infinite rank have a permutability transitive relation\",\"authors\":\"Adolfo Ballester Bolinches, Maria De Falco, Francesco de Giovanni, Carmela Musella\",\"doi\":\"10.1515/jgth-2023-0296\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n Let 𝐺 be a group.\\nA subgroup 𝐻 of 𝐺 is called permutable if \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mrow>\\n <m:mrow>\\n <m:mi>H</m:mi>\\n <m:mo>⁢</m:mo>\\n <m:mi>X</m:mi>\\n </m:mrow>\\n <m:mo>=</m:mo>\\n <m:mrow>\\n <m:mi>X</m:mi>\\n <m:mo>⁢</m:mo>\\n <m:mi>H</m:mi>\\n </m:mrow>\\n </m:mrow>\\n </m:math>\\n \\n HX=XH\\n \\n for all subgroups 𝑋 of 𝐺.\\nPermutability is not in general a transitive relation, and 𝐺 is called a \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>PT</m:mi>\\n </m:math>\\n \\n \\\\mathrm{PT}\\n \\n -group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>PT</m:mi>\\n </m:math>\\n \\n \\\\mathrm{PT}\\n \\n is not inherited by subgroups, and 𝐺 is called a \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mover accent=\\\"true\\\">\\n <m:mi>PT</m:mi>\\n <m:mo>̄</m:mo>\\n </m:mover>\\n </m:math>\\n \\n \\\\overline{\\\\mathrm{PT}}\\n \\n -group if all its subgroups have the \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>PT</m:mi>\\n </m:math>\\n \\n \\\\mathrm{PT}\\n \\n -property.\\nWe prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>PT</m:mi>\\n </m:math>\\n \\n \\\\mathrm{PT}\\n \\n -property, then 𝐺 is a \\n \\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mover accent=\\\"true\\\">\\n <m:mi>PT</m:mi>\\n <m:mo>̄</m:mo>\\n </m:mover>\\n </m:math>\\n \\n \\\\overline{\\\\mathrm{PT}}\\n \\n -group.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-17\",\"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-2023-0296\",\"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-2023-0296","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

让𝐺是一个群。如果对于𝐺的所有子群𝑋来说，H X = X HX=XH ，那么𝐺的子群𝐻就叫做可周遍群。可周遍性在一般情况下不是一个传递关系，如果，只要𝐺是𝐺的可周遍子群，并且𝐻是𝐺的可周遍子群，我们总是有𝐻在𝐺中是可周遍的，那么𝐺就叫做PT \mathrm{PT} -群。子群不会继承 PT \mathrm{PT} 的属性，如果𝐺 的所有子群都是可变子群，那么𝐺 被称为 PT ̄ \overline\mathrm{PT}} 。 -我们证明，如果𝐺是一个无穷秩的可解群，其无穷秩的适当子群具有 PT \mathrm{PT} -属性，那么𝐺就是一个 PT ̄ \overline\mathrm{PT}} -群。 -群。

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