张 丽,郭文彬,陈啸宇.p-超循环嵌入子群的一个判别准则[J].数学年刊A辑,2018,39(3):297~308
p-超循环嵌入子群的一个判别准则
A Characterization of p-Hypercyclically Embedded Subgroups of Finite Groups
Received: December 20, 2015  Revised: September 06, 2017
DOI:10.16205/j.cnki.cama.2018.0026
中文关键词:  Sylow $p${-}子群, $mathcal{U}$-$Phi${-}可补充子群,$p${-}超可解群, $p${-}幂零群
英文关键词:Sylow $p$-subgroup, $mathcal{U}$-$Phi$-Supplemented subgroup, $p$-Supersolvable group, $p$-Nilpotent group
基金项目:本文受到国家自然科学基金(No.11771409), 安徽建筑大学科研启动基金(No.K10807) 和南京师范大学科研启动基金(No.2015101XGQ0105)的资助.
Author NameAffiliationE-mail
ZHANG School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230022, China. zhangli0516@ahjzu.edu.cn 
GUO Wenb Corresponding author. School of Mathematics Sciences, University of Science and Technology of China, Hefei 230026, China. wbguo@ustc.edu.cn 
CHEN Xiao School of Mathematics Sciences, Nanjing Normal University, Nanjing 210023, China. jelly@njnu.edu.cn 
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中文摘要:
      令$E$是有限群$G$的一个正规子群, 且$\mathcal{U}$是所有有限超可解群的集合. $E$称为在$G$中是$p${-}超循环嵌入的, 如果$E$的每个$pd${-}阶的$G${-}主因子是循环的. $G$的子群$H$称为在$G$中是$\mathcal{U}$-$\Phi${-}可补充的, 如果存在$G$的一个次正规 子群$T$, 使得$G=HT$, 且$(H\cap T)H_{G}/H_{G}\leq\Phi(H/H_{G})Z_{\mathcal{U}}(G/H_{G})$, 其中$Z_{\mathcal{U}}(G/H_{G})$是商群$G/H_{G}$的$\mathcal{U}${-}超中心. 作者证明, 如果$E$的一些$p${-}子群在$G$中是$\mathcal{U}$-$\Phi${-}可补充的, 那么$E$在$G$中是$p${-}超循环嵌入的. 作为应用, 得到了有限群是$p${-}超可解的若干判断准则, 并且推广了一些已知的结果.
英文摘要:
      Let $E$ be a normal subgroup of a finite group $G$ and $\mathcal{U}$ the class of all finite supersolvable groups. $E$ is said to be $p$-hypercyclically embedded in $G$ if every $pd$-$G$-chief factor below $E$ is cyclic. A subgroup $H$ of $G$ is $\mathcal{U}$-$\Phi$-supplemented in $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G=HT$ and $(H\cap T)H_{G}/H_{G}\leq\Phi(H/H_{G})Z_{\mathcal{U}}(G/H_{G})$, where $Z_{\mathcal{U}}(G/H_{G})$ is the $\mathcal{U}$-hypercentre of $G/H_{G}$. In this paper, it is proved that $E$ is $p$-hypercyclically embedded in $G$ if some classes of $p$-subgroups of $E$ are $\mathcal{U}$-$\Phi$-supplemented in $G$. As applications, some new characterizations of $p$-supersolvability of finite groups are obtained and some recent results are extended.
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