Locally Transitive Graphs Admitting a Group with Cyclic Sylow Subgroups

All graphs are finite simple undirected and of no isolated vertices in this paper. Using the theory of coset graphs and permutation groups, it is completed that a classification of locally transitive graphs admitting a non-Abelian group with cyclic Sylow subgroups. They are either the union of the family of arc-transitive graphs, or the union of the family of bipartite edge-transitive graphs.

Finite Groups with Seminormal Sylow Subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.

A Characterization of L_n(q) by the Normalizers' Orders of Their Sylow Subgroups

Let G be a finite group. If |NG（R1）|=|NLn（q）（R2）| for every prime r, where R1∈Sylr G and R2∈Sylr（Ln（q））, then G≌Ln（q）.

Finite Groups Acting 2-Transitively on Their Sylow Subgroups

In this paper we have completely determined:(1)all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups.(2)all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T.(3)all finite groups which are 2-transitive on all their sets of Sylow subgroups.

Normal Sylow subgroups and monomial Brauer characters

Let G be a finite group,p be a prime divisor of|G|,and P be a Sylow p-subgroup of G.We prove that P is normal in a solvable group G if|G:kerφ|_(p′)=φ(1)_(p′)for every nonlinear irreducible monomial p-Brauer characterφof G,where kerφis the kernel ofφandφ(1)_(p′)is the p′-part ofφ(1).

Sub-cover-avoidance Properties and the Structure of Finite Groups

A subgroup H of a group G is said to have the sub-cover-avoidance property in G ffthereis a chief series 1 = G0 ≤ G1 ≤…≤ Gn - G, such that Gi-1（H ∩ Gi） G for every i = 1,2,... ,l. In this paper, we give some characteristic conditions for a group to be solvable under the assumptions that some subgroups of a group satisfy the sub-cover-avoidance property.

X-s-Permutable Subgroups

Let X be a nonempty subset of a group G. A subgroup H of G is said to be X-spermutable in G if, for every Sylow subgroup T of G, there exists an element x E X such that HT^x= T^xH. In this paper, we obtain some results about the X-s-permutable subgroups and use them to determine the structure of some finite groups.

c^＊-Quasinormally embedded subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally 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-quasinormal subgroup of G; H is called c^＊-quasinormally embedded in G if there is a subgroup T of G such that G = HT and HCqT is s-quasinormally embedded in G. We investigate the influence of c^＊-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.

On m-Embedded Primary Subgroups of Finite Groups

Suppose that A is a subgroup of a group G.A is called to be m-embeddedin G if G has a subnormal subgroup T and a{1<G}-embedded subgroup C such thatG=AT and AnT<C≤A.In this paper,we shall investigate the structure of finitegroups by using m-embedded subgroups and obtain some new characterization aboutp-supersolvability and generalized hypercentre of finite groups.Some results in Guoand Shum(Arch Math 80:561-569,2003),Ramadan et al.(Arch Math 85:203-210,2005),Tang and Miao(Turk J Math 39:501-506,2015),and Xu and Zhang(Can MathBull 57(4):884-889,2014)are generalized.

A New Characterization of L_2(r) by Their Sylow Numbers

Let G be a finite centerless group, let π（G） be the set of prime divisors of the order of G, and let np（G） be the number of Sylow p-subgroups of G, that is, np（G） = ｜Sylp（G）｜. Set NS（G） ：= ｜np（G）｜ p ∈ π（G）}. In this paper, we are investigating whether L2（r） is determined up to isomorphism by NS（L2（r）） when r is prime.

New Criteria for Quasi-F-Groups

Let G be a finite group and F a saturated formation of finite groups.Then G is a quasi-F-group if for every F-eccentric chief factor H/K of G and every x∈G,x induces an inner automorphism on H/K.In this article,we obtain some results about the quasi-F-groups and use them to give the conditions under which a group is quasisupersoluble.

Factorizations of Finite Groups and Related Topics

This is a survey on the recent progress in the theory of finite groups with factorizations and around it,done by the author and his coauthors,and this has no pretensions to cover all topics in this wide area of research.In particular,we only touch the great consequences of the fundamental paper of Liebeck,Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations.In each case the reader can find additional references at the end of Section 1.Some of the methods of investigation can be used to obtain information about finite groups in general,nilpotent algebras and related nearrings.