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 lohtain in Z∞（G） or NG（P） is nilpotent, arbitary p ∈ π（G）. In this pap...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 lohtain in Z∞（G） or NG（P） is nilpotent, arbitary p ∈ π（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.展开更多
基金the National Natural Science Foundation of China （No. 10571181） the Natural Science Foundation of Guangdong Province （No. 06023728）.Acknowledgement The author wishes to thank Prof. Guo Wenbin for his help. The author also thanks the referees for their helpful comments.
文摘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 lohtain in Z∞（G） or NG（P） is nilpotent, arbitary p ∈ π（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.
