{"title":"On Cyclic Actions of Finite Groups","authors":"Lü Gong, Qilin Chen, Baojun Li","doi":"10.1007/s00009-024-02626-z","DOIUrl":null,"url":null,"abstract":"<p>Let <i>A</i> act on a group <i>G</i>. <i>A</i> is said to act <i>p</i>-cyclically on <i>G</i> if <span>\\(A^{{\\mathfrak {N}}_p {\\mathfrak {A}}_{p-1}}\\)</span> acts stably on <span>\\(G/O_{p'}(G)\\)</span>, and <i>A</i> is said to act cyclically on <i>G</i> if <i>A</i> acts <i>p</i>-cyclically on <i>G</i> for all primes <i>p</i>. In this paper, the actions of a group <i>A</i> on a <i>p</i>-group or <i>p</i>-soluble group <i>G</i> are investigated and some criteria of an action to be cyclic or <i>p</i>-cyclic are obtained.</p>","PeriodicalId":49829,"journal":{"name":"Mediterranean Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-03-28","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/s00009-024-02626-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
Abstract
Let A act on a group G. A is said to act p-cyclically on G if \(A^{{\mathfrak {N}}_p {\mathfrak {A}}_{p-1}}\) acts stably on \(G/O_{p'}(G)\), and A is said to act cyclically on G if A acts p-cyclically on G for all primes p. In this paper, the actions of a group A on a p-group or p-soluble group G are investigated and some criteria of an action to be cyclic or p-cyclic are obtained.
如果 \(A^{{\mathfrak {N}}_p {math\frak {A}}_{p-1}}\) 稳定地作用于 \(G/O_{p'}(G)\)，则称 A 以 p 循环的方式作用于 G；如果对于所有素数 p，A 以 p 循环的方式作用于 G，则称 A 以循环的方式作用于 G。本文研究了群 A 在 p 群或 p 可溶群 G 上的作用，并得到了作用为循环或 p 循环的一些标准。
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.
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