{"title":"关于某些有限 p 群的休斯猜想","authors":"Mandeep Singh, Rohit Garg","doi":"10.1007/s10231-023-01421-z","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a group, <i>p</i> a prime and <span>\\(H_p(G)\\)</span> the subgroup of <i>G</i> generated by the elements of order different from <i>p</i>. In 1957, D. R. Hughes conjectured that either <span>\\(H_p(G)=1\\)</span>, <span>\\(H_p(G)=G\\)</span>, or <span>\\([G:H_p(G)]=p\\)</span>. In this paper, we prove this conjecture for finite extraspecial <i>p</i>-groups (where <span>\\(p>2\\)</span>), finite minimal non-abelian <i>p</i>-groups and finite non-abelian <i>p</i>-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian <i>p</i>-groups which guarantee the existence of the Hughes conjecture.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Hughes conjecture for some finite p-groups\",\"authors\":\"Mandeep Singh, Rohit Garg\",\"doi\":\"10.1007/s10231-023-01421-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>G</i> be a group, <i>p</i> a prime and <span>\\\\(H_p(G)\\\\)</span> the subgroup of <i>G</i> generated by the elements of order different from <i>p</i>. In 1957, D. R. Hughes conjectured that either <span>\\\\(H_p(G)=1\\\\)</span>, <span>\\\\(H_p(G)=G\\\\)</span>, or <span>\\\\([G:H_p(G)]=p\\\\)</span>. In this paper, we prove this conjecture for finite extraspecial <i>p</i>-groups (where <span>\\\\(p>2\\\\)</span>), finite minimal non-abelian <i>p</i>-groups and finite non-abelian <i>p</i>-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian <i>p</i>-groups which guarantee the existence of the Hughes conjecture.</p></div>\",\"PeriodicalId\":8265,\"journal\":{\"name\":\"Annali di Matematica Pura ed Applicata\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-01-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annali di Matematica Pura ed Applicata\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10231-023-01421-z\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-023-01421-z","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
摘要
让 G 是一个群，p 是一个素数，\(H_p(G)\) 是由与 p 不同阶的元素产生的 G 的子群。1957 年，D. R. Hughes 猜想，要么 \(H_p(G)=1\)，\(H_p(G)=G\)，要么 \([G:H_p(G)]=p\)。在本文中，我们证明了有限外特殊 p 群（其中 \(p>2\)）、有限最小非标注 p 群和具有循环最大子群的有限非标注 p 群的这一猜想。此外，我们还给出了保证休斯猜想存在的 2 代有限非阿贝尔 p 群的一些充分条件。
Let G be a group, p a prime and \(H_p(G)\) the subgroup of G generated by the elements of order different from p. In 1957, D. R. Hughes conjectured that either \(H_p(G)=1\), \(H_p(G)=G\), or \([G:H_p(G)]=p\). In this paper, we prove this conjecture for finite extraspecial p-groups (where \(p>2\)), finite minimal non-abelian p-groups and finite non-abelian p-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian p-groups which guarantee the existence of the Hughes conjecture.
期刊介绍：
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.
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