In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification...In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.展开更多
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
A subgroup H of a finite group G is said to have the semi-cover-avoiding property in G if there is a chief series of G such that H covers or avoids every chief factor of the series. In this paper, some new results are...A subgroup H of a finite group G is said to have the semi-cover-avoiding property in G if there is a chief series of G such that H covers or avoids every chief factor of the series. In this paper, some new results are obtained based on the assumption that some subgroups have the semi-cover-avoiding property in the group.展开更多
Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it p...Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it permutes with every member of З. Recently, using the Classification of Finite Simple Groups, Heliel, Li and Li proved tile following result： If the cyclic subgroups of prime order or order 4 iif p = 2） of every member of З are З-permutable subgroups in G, then G is supersolvable.In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.展开更多
As an application of the outer structure theorem of formations [1], we study, in this paper, the supersolvability of QOLT groupsHumphreys has showed that every QOLT group of odd order is supersolvable [2]. The example...As an application of the outer structure theorem of formations [1], we study, in this paper, the supersolvability of QOLT groupsHumphreys has showed that every QOLT group of odd order is supersolvable [2]. The example 84 shows that a QCLT group of even order is not necessarily supersolvable. Now we consider the supersolvability of QCLT groups of even order.The group G is a OLT group if, for every divisor of \G\, & has a subgroup of order d. A group (7 is a QOLT group if every homomorphic image of & is OLT group.展开更多
基金Research of the first author is supported by a NNSF grant of China(Grant No.11371335)Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Science.The second author was supported by the Russian Foundation for Basic Research(Project No.13-01-00469)+1 种基金the Complex Program of UB RAS(Project 15-16-1-5)under the Agreement 02.A03.21.0006 of 27.08.2013 between the Ministry of Education and Science of the Russian Federation and Ural Federal University.
文摘In this paper,we determine the finite minimal non-supersolvable groups decomposable into the product of two normal supersolvable subgroups.
文摘In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.
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(10471085)the Shanghai Pujiang Program(05PJ14046)the Special Funds for Major Specialities of Shanghai Education Committee
文摘A subgroup H of a finite group G is said to have the semi-cover-avoiding property in G if there is a chief series of G such that H covers or avoids every chief factor of the series. In this paper, some new results are obtained based on the assumption that some subgroups have the semi-cover-avoiding property in the group.
基金Project supported by NSF of China(10571181)Advanced Academic Center of ZSU
文摘Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it permutes with every member of З. Recently, using the Classification of Finite Simple Groups, Heliel, Li and Li proved tile following result： If the cyclic subgroups of prime order or order 4 iif p = 2） of every member of З are З-permutable subgroups in G, then G is supersolvable.In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.
文摘As an application of the outer structure theorem of formations [1], we study, in this paper, the supersolvability of QOLT groupsHumphreys has showed that every QOLT group of odd order is supersolvable [2]. The example 84 shows that a QCLT group of even order is not necessarily supersolvable. Now we consider the supersolvability of QCLT groups of even order.The group G is a OLT group if, for every divisor of \G\, & has a subgroup of order d. A group (7 is a QOLT group if every homomorphic image of & is OLT group.
