| On $p$-Cover-Avoid and $S$-Quasinormally Embedded Subgroups in Finite Groups |
| Received:May 27, 2008 Revised:October 06, 2008 |
| Key Words:
$p$-cover-avoid subgroup $S$-quasinormally embedded subgroup $p$-nilpotent group.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10571181), the National Natural Science Foundation of Guangdong Province (Grant No.06023728) and the Specialized Research Fund of Guangxi University (Grant No.DD051024). |
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| Abstract: |
| Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$-subgroup of $G$. If $d$ is the smallest generator number of $P$, then there exist maximal subgroups $P_1$, $P_2$,\,$\ldots$\,, $P_d$ of $P$, denoted by ${\cal M}_d(P)=\lbrace P_1,\ldots, P_d\rbrace$, such that $\bigcap_{i=1}^d P_i=\Phi(P)$, the Frattini subgroup of $P$. In this paper, we will show that if each member of some fixed ${\cal M}_d(P)$ is either $p$-cover-avoid or $S$-quasinormally embedded in $G$, then $G$ is $p$-nilpotent. As applications, some further results are obtained. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2010.04.019 |
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