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有限群的$p$-覆盖远离子群及$S$-拟正规嵌入子群
引用本文:何宣丽,王燕鸣.有限群的$p$-覆盖远离子群及$S$-拟正规嵌入子群[J].数学研究与评论,2010,30(4):743-750.
作者姓名:何宣丽  王燕鸣
作者单位:中山大学数计学院, 广州 广东 510275; 2. 广西大学数学与信息科学学院, 广西 南宁 530004;岭南大学及中山大学数计学院, 广州 广东 510275
基金项目:国家自然科学基金(Grant No.10571181), 广东省自然科学基金(Grant No.06023728),广西大学专项基金(Grant No.DD051024).
摘    要: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 P1, P2,..., Pd of P, denoted by Md(P) = {P1,...,Pd}, such that di=1 Pi = Φ(P), the Frattini subgroup of P. In this paper, we will show that if each member of some fixed Md(P) is either p-cover-avoid or S-quasinormally embedded in G, then G is p-nilpotent. As applications, some further results are obtained.

关 键 词:p-cover-avoid  subgroup  S-quasinormally  embedded  subgroup  p-nilpotent  group.
收稿时间:2008/5/27 0:00:00
修稿时间:2008/10/6 0:00:00

On $p$-Cover-Avoid and $S$-Quasinormally Embedded Subgroups in Finite Groups
Xuan Li HE and Yan Ming WANG.On $p$-Cover-Avoid and $S$-Quasinormally Embedded Subgroups in Finite Groups[J].Journal of Mathematical Research and Exposition,2010,30(4):743-750.
Authors:Xuan Li HE and Yan Ming WANG
Institution:1. Department of Mathematics,Zhongshan University,Guangdong 510275,P.R.China;College of Mathematics and Information Science,Guangxi University,Guangxi 530004,P.R.China
2. Lingnan College and Department of Mathematics,Zhongshan University,Guangdong 510275,P.R.China
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.
Keywords:$p$-cover-avoid subgroup  $S$-quasinormally embedded subgroup  $p$-nilpotent group  
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