李样明.“具有幂零局部子群的有限群”一文的注记[J].数学研究及应用,2008,28(3):609~612 |
“具有幂零局部子群的有限群”一文的注记 |
Notes on ``Finite Groups with Nilpotent Local Subgroups'' |
投稿时间:2006-09-28 修订日期:2007-03-23 |
DOI:10.3770/j.issn:1000-341X.2008.03.021 |
中文关键词: PN-子群 亚幂零群 群结构. |
英文关键词:PN-group meta-nilpotent group structure theorem. |
基金项目:国家自然科学基金(No.10571181); 广东省自然科学基金(No.06023728). |
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中文摘要: |
称有限群$G$为一个PN-群若 $G$非幂零群,且对$G$的每一个$p$-子群$P$, 或者$P$是$G$的正规子群, 或者$P \subseteq Z_\infty(G)$, 或者$N_G(P)$是幂零群, $\forall p \in \pi(G)$. 本文证明了PN-群是亚幂零群. 特别地, PN-群是可解的 且给出了PN-群结构定理的一个初等的、直观的、简洁的证明. |
英文摘要: |
A finite group $G$ is called PN-group if $G$ is not nilpotent and for every $p$-subgroup $P$ of $G$, there holds that either $P$ is normal in $G$ or $P \subseteq Z_\infty(G)$ or $N_G(P)$ is nilpotent, $\forall p \in \pi(G)$. In this paper, we prove that PN-group is meta-nilpotent, especially, PN-group is solvable. In addition, we give an elementary, intuitionistic, compact proof of the structure theorem of PN-group. |
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