Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = ...Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H N K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature.展开更多
Let P be a Sylow p-subgroup of a group G with the smallest generator number d, where p is a prime. Denote by Md（P） = （P1, P2,..., Pd） a set of maximal subgroups of P such that φ（P） = ∩n^d=1Pn. In this paper, w...Let P be a Sylow p-subgroup of a group G with the smallest generator number d, where p is a prime. Denote by Md（P） = （P1, P2,..., Pd） a set of maximal subgroups of P such that φ（P） = ∩n^d=1Pn. In this paper, we investigate the structure of a finite group G under the assumption that the maximal subgroups in Md（P） are weakly s-permutably embedded in G, some interesting results are obtained which generalize some recent results. Finally, we give some further results in terms of weakly s-permutably embedded subgroups.展开更多
In the literature, p-nilpotency of the normalizers of p-subgroups has an important influence on finite p-nilpotent groups. In this paper, we extend the p-nilpotency to psupersolvability and choose every normal p-subgr...In the literature, p-nilpotency of the normalizers of p-subgroups has an important influence on finite p-nilpotent groups. In this paper, we extend the p-nilpotency to psupersolvability and choose every normal p-subgroups H of P such that |H| = pdand explore p-supersolvability of G by the conditions of weakly M-supplemented properties of H and psupersolvability of the normalizer NG(H), where 1 ≤ pd<|P |. Also, we study the p-nilpotency of G under the assumptions that NG(P) is p-nilpotent and the weakly M-supplemented condition on a subgroup K such that K_(p)■K and P′≤ K_(p) ≤ Φ(P), Kp is a Sylow p-subgroup K. To some extent, our main results can be regarded as generalizations of the Frobenius theorem.展开更多
Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup...Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H. In this paper, we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H VIT ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. Some results about the influence of weakly S-embedded subgroups on the structure of finite groups are given.展开更多
A subgroup H of a group G is said to be weakly normal in G if Hg ≤ NG（H） implies that g E NG（H）. In this paper, using the condition that the minimal subgroups or subgroups of prime square order of G are weakly no...A subgroup H of a group G is said to be weakly normal in G if Hg ≤ NG（H） implies that g E NG（H）. In this paper, using the condition that the minimal subgroups or subgroups of prime square order of G are weakly normal in G, we get some results about formation.展开更多
We prove that a finite group G is p-supersolvable or p-nilpotent if some sub- groups of G are weakly s-semipermutable in G. Several earlier results are generalized.
Suppose that G is a finite group and H is a subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing ｜H｜, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgrou...Suppose that G is a finite group and H is a subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing ｜H｜, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup Hse of G contained in H such that G = HT and H n T ≤ Hse. In this paper, we continue the work of [Comm. Algebra, 2009, 37： 1086-1097] to study the influence of the weakly s-permutably embedded subgroups on the structure of finite groups, and we extend some recent results.展开更多
Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F a...Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.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
基金supported by the Deanship of Scientific Research(DSR) at King Abdulaziz University(KAU) represented by the Unit of Research Groups through the grant number(MG/31/01) for the group entitled "Abstract Algebra and its Applications"
文摘Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H N K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature.
基金the National Natural Science Foundation of China(Grant Nos.1120108211171353)+2 种基金China Postdoctoral Science Foundation(Grant No.2013T60866)the Natural Science Foundation of Guangdong Province(Grant No.S201204007267)Outstanding Young Teachers Training Project of Guangdong Province(GrantNo.Yq2013061)
文摘Let P be a Sylow p-subgroup of a group G with the smallest generator number d, where p is a prime. Denote by Md（P） = （P1, P2,..., Pd） a set of maximal subgroups of P such that φ（P） = ∩n^d=1Pn. In this paper, we investigate the structure of a finite group G under the assumption that the maximal subgroups in Md（P） are weakly s-permutably embedded in G, some interesting results are obtained which generalize some recent results. Finally, we give some further results in terms of weakly s-permutably embedded subgroups.
基金Supported by the National Natural Science Foundation of China(Grant No.12001436)the Natural Science Foundation of Sichuan Province(Grant No.2022NSFSC1843)+3 种基金Chunhui Plan Cooperative Scientific Research Project of Ministry of Education of the People’s Republic of Chinathe Fundamental Research Funds of China West Normal University(Grant Nos.17E09118B032)。
文摘In the literature, p-nilpotency of the normalizers of p-subgroups has an important influence on finite p-nilpotent groups. In this paper, we extend the p-nilpotency to psupersolvability and choose every normal p-subgroups H of P such that |H| = pdand explore p-supersolvability of G by the conditions of weakly M-supplemented properties of H and psupersolvability of the normalizer NG(H), where 1 ≤ pd<|P |. Also, we study the p-nilpotency of G under the assumptions that NG(P) is p-nilpotent and the weakly M-supplemented condition on a subgroup K such that K_(p)■K and P′≤ K_(p) ≤ Φ(P), Kp is a Sylow p-subgroup K. To some extent, our main results can be regarded as generalizations of the Frobenius theorem.
基金supported by National Natural Science Foundation of China (Grant Nos.10771172,11001226)Postgraduate Innovation Foundation of Southwest University (Grant Nos. ky2009013,ky2010007)
文摘Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H. In this paper, we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H VIT ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. Some results about the influence of weakly S-embedded subgroups on the structure of finite groups are given.
基金This work was supported by the National Natural Science Foundation of China （Grant No. 11171243, 11326056） and the Scientific Research Foundation for Doctors, Henan University of Science and Technology （No. 09001610）.
文摘A subgroup H of a group G is said to be weakly normal in G if Hg ≤ NG（H） implies that g E NG（H）. In this paper, using the condition that the minimal subgroups or subgroups of prime square order of G are weakly normal in G, we get some results about formation.
基金Research of the authors is supported by NNSF of China （Grants 11171243 and 11001098）, Natural Science Foundation of Jiangsu （Grant BK20140451）, and University Natural Sci- ence Foundation of Jiangsu （Grant 14KJB110002）.
文摘We prove that a finite group G is p-supersolvable or p-nilpotent if some sub- groups of G are weakly s-semipermutable in G. Several earlier results are generalized.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271085, 11201082）, the Natural Science Foundation of Guangdong Province （S2011010004447）, and the Special Project for the Subject Build of High Education of Guangdong Province （2012KJCX0081）.
文摘Suppose that G is a finite group and H is a subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing ｜H｜, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup Hse of G contained in H such that G = HT and H n T ≤ Hse. In this paper, we continue the work of [Comm. Algebra, 2009, 37： 1086-1097] to study the influence of the weakly s-permutably embedded subgroups on the structure of finite groups, and we extend some recent results.
基金the Natural Science Foundation of Chinathe Natural Science Foundation of Guangxi Autonomous Region (No.0249001)
文摘Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.
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.
