{"title":"On the Norms of p-Nilpotent Residuals of Subgroups in a Finite Group","authors":"Baoyu Zhang, Quanfu Yan, Zhencai Shen","doi":"10.1007/s00009-024-02613-4","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> be a finite group and <i>p</i> be a prime. We define <span>\\(N^{\\mathcal {N}_p*}(G)\\)</span> to be the intersection of the normalizers of the <i>p</i>-nilpotent residuals of all two-generator subgroups of <i>G</i> whose <i>p</i>-nilpotent residuals are nilpotent. We show that <span>\\(N^{\\mathcal {N}_p}(G)=N^{\\mathcal {N}_p*}(G)\\)</span>. Using the method in the present paper, we will be able to give an affirmative answer to an open problem in Shen et al. (Mediterr J Math 19:191, 2022), which also indicates that similar conclusions hold for many formations. It is also proved that <span>\\(G=N^{\\mathcal {N}_p}(G)\\)</span> if and only if every three-generator subgroup <i>H</i> of <i>G</i> satisfies <span>\\(H=N^{\\mathcal {N}_p}(H)\\)</span>. To this end, we introduce and investigate the <i>IO</i>-<span>\\(N^{\\mathcal {N}_p}\\)</span>-groups, i.e., the groups <i>G</i> such that <span>\\(G\\ne N^{\\mathcal {N}_p}(G),\\)</span> but each proper subgroup and each proper quotient of <i>G</i> equals its <i>p</i>-nilpotent norm. Moreover, new results in terms of the <i>p</i>-nilpotent norm and the <i>p</i>-nilpotent hypernorm <span>\\(N^{\\mathcal {N}_p}_\\infty (G)\\)</span> are given.</p>","PeriodicalId":49829,"journal":{"name":"Mediterranean Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mediterranean Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02613-4","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a finite group and p be a prime. We define \(N^{\mathcal {N}_p*}(G)\) to be the intersection of the normalizers of the p-nilpotent residuals of all two-generator subgroups of G whose p-nilpotent residuals are nilpotent. We show that \(N^{\mathcal {N}_p}(G)=N^{\mathcal {N}_p*}(G)\). Using the method in the present paper, we will be able to give an affirmative answer to an open problem in Shen et al. (Mediterr J Math 19:191, 2022), which also indicates that similar conclusions hold for many formations. It is also proved that \(G=N^{\mathcal {N}_p}(G)\) if and only if every three-generator subgroup H of G satisfies \(H=N^{\mathcal {N}_p}(H)\). To this end, we introduce and investigate the IO-\(N^{\mathcal {N}_p}\)-groups, i.e., the groups G such that \(G\ne N^{\mathcal {N}_p}(G),\) but each proper subgroup and each proper quotient of G equals its p-nilpotent norm. Moreover, new results in terms of the p-nilpotent norm and the p-nilpotent hypernorm \(N^{\mathcal {N}_p}_\infty (G)\) are given.
期刊介绍:
The Mediterranean Journal of Mathematics (MedJM) is a publication issued by the Department of Mathematics of the University of Bari. The new journal replaces the Conferenze del Seminario di Matematica dell’Università di Bari which has been in publication from 1954 until 2003.
The Mediterranean Journal of Mathematics aims to publish original and high-quality peer-reviewed papers containing significant results across all fields of mathematics. The submitted papers should be of medium length (not to exceed 20 printed pages), well-written and appealing to a broad mathematical audience.
In particular, the Mediterranean Journal of Mathematics intends to offer mathematicians from the Mediterranean countries a particular opportunity to circulate the results of their researches in a common journal. Through such a new cultural and scientific stimulus the journal aims to contribute to further integration amongst Mediterranean universities, though it is open to contribution from mathematicians across the world.