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数学进展 - 2020, Vol. 49(4): 385-400
综述文章
有限群的半置换子群与${s}$-半置换子群
Semipermutable Subgroups and ${s}$-semipermutable Subgroups in Finite Groups

李样明
LI Yangming

广东第二师范学院数学系, 广州, 广东, 510310
Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong, 510310, P. R. China

收稿日期: 2019-08-15
出版日期: 2020-08-11
2020, Vol. 49(4): 385-400
DOI: 10.11845/sxjz.2019008a


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摘要 设$H$为$G$的子群, 称$H$在$G$中半置换, 如果$HK = KH$对任意满足条件$(|H|, |K|) =1$ 的$G$的子群$K$ 都成立; 称$H$ 在$G$ 中$s$- 半置换, 如果$HP=PH$对任意$P\in \mathrm{Syl}_p(G)$ 都成立, 其中$(|H|,p)=1$. 这两个概念自陈重穆1987年提出后, 获得国内外许多学者的关注, 应用此概念近几十年来有大量的文章出现. 本文对这方面的成果进行总结, 给出研究过程中的思路.
关键词 半置换子群$s$-半置换子群极大子群极小子群广义Fitting子群群系    
Abstract:Suppose that $H$ is a subgroup of a finite group $G$. We call $H$ is semipermutable in $G$ if $HK = KH$ for any subgroup $K$ of $G$ such that $(|H|, |K|) =1$; we call $H$ is $s$-semipermutable in $G$ if $HG_p = G_pH$, for any Sylow $p$-subgroup $G_p$ of $G$ such that $(|H|, p) =1$. These two concepts have been received attention of many scholars in group theory since they were introduced by Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and give some thoughts in the research process.
Key wordssemipermutable subgroups    $s$-semipermutable subgroups    maximal subgroups    minimal subgroups    generalized Fitting-subgroup    formation
PACS:  O152.2  
基金资助:广东省基础研究及应用研究重大项目(自然科学) (No. 2017KZDXM058)和广州市科技计划项目 (No. 201804010088).
通讯作者: E-mail: liyangming@gdei.edu.cn   
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