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
The chemical formula of omphacite was expressed with （M<sub>Ⅱ</sub>M<sub>Ⅰ</sub>）（Si, AI）<sub>2</sub>O<sub>6</sub>.Cations that occupied the M<sub>Ⅱ</sub...The chemical formula of omphacite was expressed with （M<sub>Ⅱ</sub>M<sub>Ⅰ</sub>）（Si, AI）<sub>2</sub>O<sub>6</sub>.Cations that occupied the M<sub>Ⅱ</sub> site were large cations such as Ca or Na, and Na/（Na+Ca） ratio ranged from 0.2 to 0.8; the 6-coordinated M<sub>Ⅰ</sub> site accommodated cations such as Mg, Fe<sup>2+</sup>,Al, Fe<sup>3+</sup> ,and Al/（Al+Fe<sup>3</sup>+） ratio was more than 0.5 Omphacite space groups reported were C2/c, P2, P2/n, P2/c, cell parameters are a<sub>0</sub> = 0.9600—0.9630 nm, b<sub>0</sub> = 0.8750—0.8830 nm, C<sub>0</sub> =0.5230—0.5290 nm,β = 106°40’—107°10’. The sample was picked up from the eclogites in Zhucheng County, Shangdong Province. The intensity data were collected with the RIGAKU RASA Ⅱ -S four-circle single crystal diffractometer. The correct structure model was obtained by using the Patterson method and Fourier synthesis, SHELX-76 program, structure refinement with 905 independent diffraction points （|F<sub>0</sub>|】 3σ|F<sub>0</sub> |）. After the structure parameter refinement, the R-factor reduced to 0.0515. The crystal structure analysis indicates that omphacite has a new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.
文摘The chemical formula of omphacite was expressed with （M<sub>Ⅱ</sub>M<sub>Ⅰ</sub>）（Si, AI）<sub>2</sub>O<sub>6</sub>.Cations that occupied the M<sub>Ⅱ</sub> site were large cations such as Ca or Na, and Na/（Na+Ca） ratio ranged from 0.2 to 0.8; the 6-coordinated M<sub>Ⅰ</sub> site accommodated cations such as Mg, Fe<sup>2+</sup>,Al, Fe<sup>3+</sup> ,and Al/（Al+Fe<sup>3</sup>+） ratio was more than 0.5 Omphacite space groups reported were C2/c, P2, P2/n, P2/c, cell parameters are a<sub>0</sub> = 0.9600—0.9630 nm, b<sub>0</sub> = 0.8750—0.8830 nm, C<sub>0</sub> =0.5230—0.5290 nm,β = 106°40’—107°10’. The sample was picked up from the eclogites in Zhucheng County, Shangdong Province. The intensity data were collected with the RIGAKU RASA Ⅱ -S four-circle single crystal diffractometer. The correct structure model was obtained by using the Patterson method and Fourier synthesis, SHELX-76 program, structure refinement with 905 independent diffraction points （|F<sub>0</sub>|】 3σ|F<sub>0</sub> |）. After the structure parameter refinement, the R-factor reduced to 0.0515. The crystal structure analysis indicates that omphacite has a new
