{"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}
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.
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