Some Criteriafor p-SupersolvabilityofaFiniteGroup

A subgroup H of a finite group G is said to be S-semipermutable in G if H permutes with all Sylow q-subgroups of G for the primes q not dividing the order of H.Some criteria for p-supersolvability of a finite group are given,which are the generalizations of many recent results.

The Action on p-Groups and p-Supersolvability

Let a finite group A act on a finite p-group P with ｜P｜〉 p^e, where e is an integer with e 〉 2. In this paper, we show that P is centralized by OP（A^Ap-1） if every non-maximal class p-group of order p^e in P is stabilized by OB（A）. As applications, someconditions are given for a finite group to be p-supersolvable.

Notes on Solvability and p-Supersolvability of Finite Groups

A subgroup H of G is called M-supplemented in G if there exists a subgroup B of G such that G = HB and HiB < G for every maximal subgroup Hi of H. In this paper, we use M-suppiemented subgroups to study the structure of finite groups and obtain some new characterization about solvability and p-supersolvability for a fixed prime p. Some results in the literature are corollaries of our theorems.

s-Permutably embedded subgroups and p-supersolvable groups

Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.

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.

On c#-normal subgroups in finite groups

A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H n K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c#- normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.

Onc*-Normal Subgroups in Finite Groups

A subgroup H of a finitegroup G is called a c＊-normal subgroup of G if there exists a

On Weakly s-Semipermutable Subgroups of Finite Groups

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.

Local Covering Subgroups in Finite Groups

A subgroup A of a finite group G is called a local covering subgroup of G if A^(G)=AB for all maximal G-invariant subgroup B of A^(G)=(A^(G),g∈G).Let p be a prime and d be a positive integer.Assume that all subgroups of p^(d),and all cyclic subgroups of order 4 when p^(d)=2 and a Sylow2-subgroup of G is nonabelian,of G are local covering subgroups.Then G is p-supersolvable whenever p^(d)=p or p^(d)≤(√|G|_(p))or p^(d)≤|O_(p'p)(G)|_(p)/p.