| 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. |
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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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