{"title":"具有亚正态或模态施密特𝑝-𝑑-子群的群","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":50188,"journal":{"name":"Journal of Group Theory","volume":"110 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":50188,\"journal\":{\"name\":\"Journal of Group Theory\",\"volume\":\"110 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-02-09\",\"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-2023-0096\",\"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-2023-0096","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
施密特群是一个有限非零能群，它的每个适当子群都是零能的。在本文中，我们证明了如果有限群𝐺 的每个施密特子群都是子常群或模群，那么 G / F ( G ) G/F(G) 是循环群。此外，对于给定素数𝑝，我们描述了具有阶可被𝑝整除的亚正态或模态施密特子群的有限群的结构。
Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups
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.
期刊介绍：
The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.
Topics:
Group Theory-
Representation Theory of Groups-
Computational Aspects of Group Theory-
Combinatorics and Graph Theory-
Algebra and Number Theory
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