Notes on ``Finite Groups with Nilpotent Local Subgroups'' |
Received:September 28, 2006 Revised:March 23, 2007 |
Key Words:
PN-group meta-nilpotent group structure theorem.
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Fund Project:the National Natural Science Foundation of China (No.10571181); the Natural Science Foundation of Guangdong Province (No.06023728). |
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Abstract: |
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. |
Citation: |
DOI:10.3770/j.issn:1000-341X.2008.03.021 |
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