{"title":"论有限群中的次模性和 K $$mathfrak{F}$ -次规范性","authors":"V. S. Monakhov, I. L. Sokhor","doi":"10.1134/s0081543823060159","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(\\mathfrak{F}\\)</span> be a formation, and let <span>\\(G\\)</span> be a finite group. A subgroup <span>\\(H\\)</span> of <span>\\(G\\)</span> is called <span>\\(\\mathrm{K}\\mathfrak{F}\\)</span>-subnormal (submodular) in <span>\\(G\\)</span> if there is a subgroup chain <span>\\(H=H_{0}\\leq H_{1}\\leq\\mathinner{\\ldotp\\ldotp\\ldotp}\\leq H_{n-1}\\leq H_{n}=G\\)</span> such that, for every <span>\\(i\\)</span> either <span>\\(H_{i}\\)</span> is normal in <span>\\(H_{i+1}\\)</span> or <span>\\(H_{i+1}^{\\mathfrak{F}}\\leq H_{i}\\)</span> (<span>\\(H_{i}\\)</span> is a modular subgroup of <span>\\(H_{i+1}\\)</span>, respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is <span>\\(\\mathrm{K}\\mathfrak{U}_{1}\\)</span>-subnormal. Here <span>\\(\\mathfrak{U}_{1}\\)</span> is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation <span>\\(\\mathfrak{F}\\)</span>, every solvable <span>\\(\\mathrm{K}\\mathfrak{F}\\)</span>-subnormal subgroup of a group <span>\\(G\\)</span> is contained in the solvable radical of <span>\\(G\\)</span>. We also obtain a series of applications of these results to the investigation of groups factorized by <span>\\(\\mathrm{K}\\mathfrak{F}\\)</span>-subnormal and submodular subgroups.\n</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"6 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Submodularity and K $$\\\\mathfrak{F}$$ -Subnormality in Finite Groups\",\"authors\":\"V. S. Monakhov, I. L. Sokhor\",\"doi\":\"10.1134/s0081543823060159\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\(\\\\mathfrak{F}\\\\)</span> be a formation, and let <span>\\\\(G\\\\)</span> be a finite group. A subgroup <span>\\\\(H\\\\)</span> of <span>\\\\(G\\\\)</span> is called <span>\\\\(\\\\mathrm{K}\\\\mathfrak{F}\\\\)</span>-subnormal (submodular) in <span>\\\\(G\\\\)</span> if there is a subgroup chain <span>\\\\(H=H_{0}\\\\leq H_{1}\\\\leq\\\\mathinner{\\\\ldotp\\\\ldotp\\\\ldotp}\\\\leq H_{n-1}\\\\leq H_{n}=G\\\\)</span> such that, for every <span>\\\\(i\\\\)</span> either <span>\\\\(H_{i}\\\\)</span> is normal in <span>\\\\(H_{i+1}\\\\)</span> or <span>\\\\(H_{i+1}^{\\\\mathfrak{F}}\\\\leq H_{i}\\\\)</span> (<span>\\\\(H_{i}\\\\)</span> is a modular subgroup of <span>\\\\(H_{i+1}\\\\)</span>, respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is <span>\\\\(\\\\mathrm{K}\\\\mathfrak{U}_{1}\\\\)</span>-subnormal. Here <span>\\\\(\\\\mathfrak{U}_{1}\\\\)</span> is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation <span>\\\\(\\\\mathfrak{F}\\\\)</span>, every solvable <span>\\\\(\\\\mathrm{K}\\\\mathfrak{F}\\\\)</span>-subnormal subgroup of a group <span>\\\\(G\\\\)</span> is contained in the solvable radical of <span>\\\\(G\\\\)</span>. We also obtain a series of applications of these results to the investigation of groups factorized by <span>\\\\(\\\\mathrm{K}\\\\mathfrak{F}\\\\)</span>-subnormal and submodular subgroups.\\n</p>\",\"PeriodicalId\":54557,\"journal\":{\"name\":\"Proceedings of the Steklov Institute of Mathematics\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-02-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Steklov Institute of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1134/s0081543823060159\",\"RegionNum\":4,\"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":"Proceedings of the Steklov Institute of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0081543823060159","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
On Submodularity and K $$\mathfrak{F}$$ -Subnormality in Finite Groups
Let \(\mathfrak{F}\) be a formation, and let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called \(\mathrm{K}\mathfrak{F}\)-subnormal (submodular) in \(G\) if there is a subgroup chain \(H=H_{0}\leq H_{1}\leq\mathinner{\ldotp\ldotp\ldotp}\leq H_{n-1}\leq H_{n}=G\) such that, for every \(i\) either \(H_{i}\) is normal in \(H_{i+1}\) or \(H_{i+1}^{\mathfrak{F}}\leq H_{i}\) (\(H_{i}\) is a modular subgroup of \(H_{i+1}\), respectively). We prove that, in a group, a primary subgroup is submodular if and only if it is \(\mathrm{K}\mathfrak{U}_{1}\)-subnormal. Here \(\mathfrak{U}_{1}\) is a formation of all supersolvable groups of square-free exponent. Moreover, for a solvable subgroup-closed formation \(\mathfrak{F}\), every solvable \(\mathrm{K}\mathfrak{F}\)-subnormal subgroup of a group \(G\) is contained in the solvable radical of \(G\). We also obtain a series of applications of these results to the investigation of groups factorized by \(\mathrm{K}\mathfrak{F}\)-subnormal and submodular subgroups.
期刊介绍:
Proceedings of the Steklov Institute of Mathematics is a cover-to-cover translation of the Trudy Matematicheskogo Instituta imeni V.A. Steklova of the Russian Academy of Sciences. Each issue ordinarily contains either one book-length article or a collection of articles pertaining to the same topic.