其正常子群之外的元素具有素幂级数的有限群的结构

IF 1.3 3区 数学 Q1 MATHEMATICS
Changguo Shao, Qinhui Jiang
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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":"17 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"17 1\",\"pages\":\"\"},\"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}","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

摘要

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