The structure of finite groups whose elements outside a normal subgroup have prime power orders

Changguo Shao, Qinhui Jiang
{"title":"The structure of finite groups whose elements outside a normal subgroup have prime power orders","authors":"Changguo Shao, Qinhui Jiang","doi":"10.1017/prm.2024.71","DOIUrl":null,"url":null,"abstract":"The structure of groups in which every element has prime power order (CP-groups) is extensively studied. We first investigate the properties of group <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline3.png\"/> </jats:alternatives> </jats:inline-formula> such that each element of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G\\setminus N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline30a.png\"/> </jats:alternatives> </jats:inline-formula> has prime power order. It is proved that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline10a.png\"/> </jats:alternatives> </jats:inline-formula> is solvable or every non-solvable chief factor <jats:inline-formula> <jats:alternatives> <jats:tex-math>$H/K$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline10.png\"/> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline11.png\"/> </jats:alternatives> </jats:inline-formula> satisfying <jats:inline-formula> <jats:alternatives> <jats:tex-math>$H\\leq N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline12.png\"/> </jats:alternatives> </jats:inline-formula> is isomorphic to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$PSL_2(3^f)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline13.png\"/> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:tex-math>$f$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline14.png\"/> </jats:alternatives> </jats:inline-formula> a 2-power. This partially answers the question proposed by Lewis in 2023, asking whether <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G\\cong M_{10}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline15.png\"/> </jats:alternatives> </jats:inline-formula>? Furthermore, we prove that if each element <jats:inline-formula> <jats:alternatives> <jats:tex-math>$x\\in G\\backslash N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline16.png\"/> </jats:alternatives> </jats:inline-formula> has prime power order and <jats:inline-formula> <jats:alternatives> <jats:tex-math>${\\bf C}_G(x)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline17.png\"/> </jats:alternatives> </jats:inline-formula> is maximal in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline18.png\"/> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline19.png\"/> </jats:alternatives> </jats:inline-formula> is solvable. Relying on this, we give the structure of group <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline20.png\"/> </jats:alternatives> </jats:inline-formula> with normal subgroup <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline21.png\"/> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:tex-math>${\\bf C}_G(x)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline22.png\"/> </jats:alternatives> </jats:inline-formula> is maximal in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline23.png\"/> </jats:alternatives> </jats:inline-formula> for any element <jats:inline-formula> <jats:alternatives> <jats:tex-math>$x\\in G\\setminus N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline24.png\"/> </jats:alternatives> </jats:inline-formula>. Finally, we investigate the structure of a normal subgroup <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline25.png\"/> </jats:alternatives> </jats:inline-formula> when the centralizer <jats:inline-formula> <jats:alternatives> <jats:tex-math>${\\bf C}_G(x)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline26.png\"/> </jats:alternatives> </jats:inline-formula> is maximal in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline27.png\"/> </jats:alternatives> </jats:inline-formula> for any element <jats:inline-formula> <jats:alternatives> <jats:tex-math>$x\\in N\\setminus {\\bf Z}(N)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline28.png\"/> </jats:alternatives> </jats:inline-formula>, which is a generalization of results of Zhao, Chen, and Guo in 2020, investigating a special case that <jats:inline-formula> <jats:alternatives> <jats:tex-math>$N=G$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000714_inline29.png\"/> </jats:alternatives> </jats:inline-formula> for our main result. We also provide a new proof for Zhao, Chen, and Guo's results above.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/prm.2024.71","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

The structure of groups in which every element has prime power order (CP-groups) is extensively studied. We first investigate the properties of group $G$ such that each element of $G\setminus N$ has prime power order. It is proved that $N$ is solvable or every non-solvable chief factor $H/K$ of $G$ satisfying $H\leq N$ is isomorphic to $PSL_2(3^f)$ with $f$ a 2-power. This partially answers the question proposed by Lewis in 2023, asking whether $G\cong M_{10}$ ? Furthermore, we prove that if each element $x\in G\backslash N$ has prime power order and ${\bf C}_G(x)$ is maximal in $G$ , then $N$ is solvable. Relying on this, we give the structure of group $G$ with normal subgroup $N$ such that ${\bf C}_G(x)$ is maximal in $G$ for any element $x\in G\setminus N$ . Finally, we investigate the structure of a normal subgroup $N$ when the centralizer ${\bf C}_G(x)$ is maximal in $G$ for any element $x\in N\setminus {\bf Z}(N)$ , which is a generalization of results of Zhao, Chen, and Guo in 2020, investigating a special case that $N=G$ for our main result. We also provide a new proof for Zhao, Chen, and Guo's results above.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
其正常子群之外的元素具有素幂级数的有限群的结构
对每个元素都有素幂级数的群(CP 群)的结构进行了广泛的研究。我们首先研究了 $G$ 群的性质,即 $G\setminus N$ 的每个元素都具有素幂阶。我们证明了 $N$ 是可解的,或者满足 $H\leq N$ 的 $G$ 的每个不可解的主因子 $H/K$ 与 $PSL_2(3^f)$ 同构,其中 $f$ 是一个 2 次幂。这部分回答了刘易斯在 2023 年提出的问题,即 $G\cong M_{10}$ 是否与 $PSL_2(3^f) $ 同构?此外,我们还证明,如果 Gbackslash N$ 中的每个元素 $x\ 都具有素幂级数,并且 ${bf C}_G(x)$ 在 $G$ 中是最大的,那么 $N$ 是可解的。基于这一点,我们给出了具有正常子群 $N$ 的组 $G$ 的结构,即对于任何元素 $x\in Gsetminus N$,${\bf C}_G(x)$ 在 $G$ 中都是最大的。最后,我们研究了当中心化${\bf C}_G(x)$ 在$G$中对任意元素$x\in N\setminus {\bf Z}(N)$ 为最大时的正则子群$N$的结构,这是对赵,陈,郭在2020年的结果的推广,研究了我们主要结果的一个特殊情况,即$N=G$。我们还为赵,陈,郭的上述结果提供了新的证明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
3.00
自引率
0.00%
发文量
72
审稿时长
6-12 weeks
期刊介绍: A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations. An international journal, publishing six issues per year, Proceedings A has been publishing the highest-quality mathematical research since 1884. Recent issues have included a wealth of key contributors and considered research papers.
期刊最新文献
The structure of finite groups whose elements outside a normal subgroup have prime power orders A unified characterization of convolution coefficients in nonlocal differential equations On a supersonic-sonic patch arising from the two-dimensional Riemann problem of the compressible Euler equations Dual formulation of constrained solutions of the multi-state Choquard equation Duality pairs, phantom maps, and definability in triangulated categories
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1