A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is p...A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups （non-PRN-group all of whose proper subgroups are PRN-groups）. At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.展开更多
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Supported by Natural Science Foundation of China （Grant No. 10871032）, Graduate Student Research and Innovation Program of Jiangsu Province （Grant No. CX10B-028Z）
文摘A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups （non-PRN-group all of whose proper subgroups are PRN-groups）. At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.
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.
