{"title":"Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups","authors":"Victor S. Monakhov, Irina L. Sokhor","doi":"10.1515/jgth-2023-0096","DOIUrl":null,"url":null,"abstract":"A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent. In this paper, we prove that if every Schmidt subgroup of a finite group 𝐺 is subnormal or modular, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>F</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</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-2023-0096_ineq_0001.png\" /> <jats:tex-math>G/F(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is cyclic. Moreover, for a given prime 𝑝, we describe the structure of finite groups with subnormal or modular Schmidt subgroups of order divisible by 𝑝.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","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-2023-0096","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent. In this paper, we prove that if every Schmidt subgroup of a finite group 𝐺 is subnormal or modular, then G/F(G)G/F(G) is cyclic. Moreover, for a given prime 𝑝, we describe the structure of finite groups with subnormal or modular Schmidt subgroups of order divisible by 𝑝.
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