设H是有限群G的一个子群,H在G中是弱Φ-可补的,如果存在G的一个子群K,使得G=HK且H∩K≤Φ(H),其中Φ(H)是H的Frattini子群.利用p阶和p^2阶子群的弱Φ-可补性,得到如下结论:1)设G是有限群,p是|G|的满足(|G|,p-1)=1的素因数.设E是G的一个...设H是有限群G的一个子群,H在G中是弱Φ-可补的,如果存在G的一个子群K,使得G=HK且H∩K≤Φ(H),其中Φ(H)是H的Frattini子群.利用p阶和p^2阶子群的弱Φ-可补性,得到如下结论:1)设G是有限群,p是|G|的满足(|G|,p-1)=1的素因数.设E是G的一个正规子群使得G/E是p-幂零群.若E p∩G N p的每个阶为p或4循环子群均在G中弱Φ-可补,那么G是p-幂零群.2)设G有限群,p是|G|满足(|G|,p^2-1)=1的素因数.设E是G的正规子群使得G/E是p-幂零的.若E p∩G N p的每个阶为p^2的子群均在G中弱Φ-可补,则G是p-幂零的.由这些结论,得到了一系列推论,推广了已知结果.展开更多
The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left...The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.展开更多
By using pronormal minimal subgroups and weak left Engel elements ofprime order of the normalizers of Sylow subgroups of a finite group G, we obtain somesufficient conditions for G to be p-nilpotent, nilpotent and sup...By using pronormal minimal subgroups and weak left Engel elements ofprime order of the normalizers of Sylow subgroups of a finite group G, we obtain somesufficient conditions for G to be p-nilpotent, nilpotent and supersolvable respectively,which generalize some known results.展开更多
文摘设H是有限群G的一个子群,H在G中是弱Φ-可补的,如果存在G的一个子群K,使得G=HK且H∩K≤Φ(H),其中Φ(H)是H的Frattini子群.利用p阶和p^2阶子群的弱Φ-可补性,得到如下结论:1)设G是有限群,p是|G|的满足(|G|,p-1)=1的素因数.设E是G的一个正规子群使得G/E是p-幂零群.若E p∩G N p的每个阶为p或4循环子群均在G中弱Φ-可补,那么G是p-幂零群.2)设G有限群,p是|G|满足(|G|,p^2-1)=1的素因数.设E是G的正规子群使得G/E是p-幂零的.若E p∩G N p的每个阶为p^2的子群均在G中弱Φ-可补,则G是p-幂零的.由这些结论,得到了一系列推论,推广了已知结果.
基金The National Natural Science Foundation of China(No.10871042)the Natural Science Foundation of Jiangsu Province(No.BK2009258)
文摘The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.
文摘By using pronormal minimal subgroups and weak left Engel elements ofprime order of the normalizers of Sylow subgroups of a finite group G, we obtain somesufficient conditions for G to be p-nilpotent, nilpotent and supersolvable respectively,which generalize some known results.
