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.  
Fund Project:the National Natural Science Foundation of China (No.10571181); the Natural Science Foundation of Guangdong Province (No.06023728).
Author NameAffiliation
LI Yang Ming Department of Mathematics, Guangdong College of Education, Guangdong 510310, China 
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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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