Let G be a finite group.A subgroup H of a group G is said to be S-quasinormal in G if H permutes with every Sylow subgroup of G.In this paper,in terms of S-quasinormal subgroups,some sufficient conditions for a finite group to be supersolvable or nilpotent are obtained.Under the condition that 2-maximal subgroup of G is S-quasinormal in G,a complete classification of G is obtained.

Let G be a finite group.A subgroup H of G is said to be π-quasinormally embedded in G,if for each prime divisor p of the order of H,a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroup of G.In terms of the properties of π-quasinormally embedded of maximal or minimal subgroups,some sufficient conditions of a saturated formation containing the class of all finite supersolvable groups or nilpotent groups are obtained,1)Let be a satuated formation containing supersolvable formation and N be a nomal subgroup of a finite group G such that GN ∈.If each maximal subgroup of any sylow subgroup of Q of F(G)of odd order is π-quasinormally embedded in NG(Q)and each maximal subgroup of a sylow 2-π subgroup of F(N)is π-quasinormally embedded in G,then G∈F.2)Let be a satuated formation containing nilpotent formation and H be a nomal subgroup of a finite group G such that GH ∈F.If each minmal subgroup or cyclic subgroup of order 4 of H is π-quasinormally embedded in G,then G∈F.And some known results are generalized and improved.

A subgroup H of a group C is said to be weakly C-normal if there exists a subnormal group K of G such that G=HK and H∩K≤H(subscript G), where H(subscript G) is the maximal normal subgroup of G contained in H. In this paper, in terms of weakly C-normality of subgroups, some sufficient conditions for a finite group to be p-nilpotent are obtained.

The main object of this paper is to show the following generalization of a Buckley's theorem and Asaad's theorem:Let H be a normal subgroup of a finite group G such that G/H is suppersolvable. If 〈x〉 is pronormal in NG(P) for every element x of P∩Gp-N with order p or 4(when p=2), where p is any prime divisor of |H|, P is a Sylow p-subgroup of H and Gp-N is the p-nilpotent residual of G, then G is supersolvable.

A finite group G is a ST-group, if, for subgroups H≤K≤L with H s-seminormal in K and K s-seminormal in L, it is always the case that H is s-seminormal in L. It is shown that for finite groups containing abnormal subgroups of prime power order, the class of solvable ST-groups coincides with the class of solvable PT-groups and also coincides with the class of solvable T-groups.

The main object of this paper is to show the following criterion for 2-nilpotence of finite groups:Let P be a sylow 2-subgroup of a finite group G. If <x>is normal in NG(P)for every element x of P∩G2-N with order 2 or 4.where G2-N is the 2-nilpetent residual of G,then G is 2-nilpotent.From the proof of the main result,the following result holds:Let P∈Syl2(G).If NG(P)is 2-nilpetent and<x>△P for all x∈P(P∩G2-N),then G is 2-nilpotent.

A subgroup H of the group G is said to be S-normal if there exists a subnormal group K of G such that G=HK and H∩K≤HSG,where HSG is the maximal subnormal subgroup of G which is contained in H.In this paper the authors give some sufficient conditions under which a group is solvable or p-nilpotent by using the S-normality of maximal subgroups,Sylow subgroups,maximal subgroup of Sylow subgroups,and second maximal subgroup of Sylow subgroups.

The purpose of this paper is to investigate the influence of subnormal subgroup on finite groups.We obtain some sufficient conditions of solvable finite groups and show the classification theorem of the finite groups whose 3-maximal subgroup are subnormal.Let G be a finite group,then all 3-maximal subgroups of G are subnormal in G if and only if G is one of the flowing two types of finite groups:(1) G is nilpotent;(2) G has a maximal subgroup M and one of the following holds:(i) M is nilpotent;(ii) M is a p-basic group of order pαq,that is,M is an inner nilpotent group whose Sylowp-subgroup is normal.

Let G be a finite group,a subgroup H of a group G is called weakly c-nornal if there exists a subnormal subgroup K of G such that G=HK and H∩K≤HG=∩g∈GHg,where HG is the largest normal subgroup of G contained in H.In this paper,in terms of weakly c-normal subgroups,some sufficient conditions for a finite group to be supersolvable or nilpotent are obtained.And some known results are generalized.

The main object of this paper is to show the following generalization of Theompson's theorem:Let M be a nilpotent maximal subgroup of a finite group G and let P be a Sylow 2-subgroup of M.If 〈x〉 is pronormal in P for all elements x of P∩G~(2-N) with order 2 or 4,where G~(2-N) is the 2-nilpotent residual of G,then G is solvable.

The main object of this paper is to show the following theorem:If every maximal subgroup of a Sylow p-subgroup Fp, let G be a finite groop andF*(G) the generalized Fitting subgroup of G, F*(G) is pronormal in NG(Fp) for all primes p∈π(F*(G)), and all the cyclic subgroups with order 2 or 4 of a Sylow 2-subgroup F2 of F*(G) are pronomal in NG(F2), then G is supersolvable.

In terms of weak left Engel element and S-seminormal subgroup, some sufficient conditions are given for a finite group to be p-nilpotent or nilpotent or supersolvable.

The main object of this paper is to show the following two theorems:(1) Let N be a normal subgroup of a finite group G such that G/N is p-nilpotent. If every dement of P∩Gp-N with order p or 4 (when p = 2) lies in Z(NG(P)) , where P ∈ Sylp(N) and Gp-N is the p-nilpotent residual of G,then G is p-nilpotent.(2) Let H be a normal subgorup of a finite group G such that G/H is nilpotent. If, for all prime divisors p of |H| and P ∈ Sylp (H), x is a weak left Engle element of Nc(P) for each element x of P ∩ Gp-N with order p or 4 (when p ＝2), where Gp-N is the p-nilpotent residual of G, then G is nilpotent.

A subgorup K of a finite gorup G is said to be π-weak quasinormal in G if K permutes with all Sylow π-subgroups of G(J Sichuan Normal University:Natural Science,2002,25(4):441-444).In this paper some more properties of π-weak quasinormal subgroups of finite groups are futher studied.In particular,some sufficient conditions and necessary conditions of π-weak quasinormal subgroups of finite groups are provided.Finally,two sufficient conditions of π-closed group are proved as follows.Let H be a Sylow π-subgroup of a finite groups G,then:(1) if N_G(H) is π-weak quasinormal in G,then G is π-closed;(2) suppose that K is a subgroup of G which contains H and is contained in(N_G(H)) and K is π-weak quasinormal in G, then G is π-closed.

有限群G的一个子群K称为G的一个π-弱拟正规子群,如果K同G的所有Sylow π-子群相乘可换(四川师范大学学报:自然科学版,2002,25(4):441-444).进一步讨论了π-弱拟正规子群的一些性质,特别是π-弱拟正规子群的一些充分条件与必要条件.最后,给出了π-闭群的两个充分条件:假设H是有限群G的一个Sylow π-子群,则:(1)如果NG(H)是G的一个π-弱拟正规子群,那么G是一个π-闭群;(2)如果M是G的一个π-弱拟正规子群并且M包含H而且M包含于NG(H),那么G是一个π-闭群.

A subgroup H of a finite group G is called s-seminormal in G if H permutes with all Sylow subgroups of G with order prime to ｜H｜. In this paper, we study some basic properties of s-seminormal subgroups of finite groups and how s-seminormal subgroups of a finite group influence the structure of the group. The main results as follows:(1)Let N be a normal subgroup of a finite group G such that G/N is p-nilpotent for some prime divisor p of |G| with (|G|,p-1)=1. If all maximal subgorups of a Sylow p-subgroup of N are s-seminormal in G, then G is p-nilpotent. (2)Let N be a normal subgroup of G such that G/N is super solvable. If all maximal subgroups of Sylow subgroups of N are s-seminormal in G, then G is supersolvable.

A subgroup H of a finite group G is called s-seminormal in G if H permutes with all Sylow subgroups of G with order prime to |H| . In this paper, the structure of finite groups G with maximal subgroups of Sylow subgroups sseminormal in G is determined: G is a group with s-seminormal Sylow subgroups and s-seminormal maximal subgroups of Sylow subgroups if and only if all maximal subgroups of Sylow subgroups of G with coprime order permute each other and each chief factor H/R of G is of prime order, moreover, if |H/R| = p then|G/CG(H/R)|= qb, where q is a prime and qb divides p - 1.

A subgroup H is called S-seminormal in a finite group G if H permutes with all Sylow p-subgroups of G with (p, |H|) = 1. The main object of this paper is to generalize some known results about finite supersolvable groups to a saturated formation containing the class of finite supersolvable groups.

By using pronormal minimal subgroups and weak left Engel elements of prime order of the normalizers of Sylow subgroups of a finite group G, we obtain some sufficient conditions for G to be p-nilpotent, nilpotent and supersolvable respectively,which generalize some known results.

In this paper, some more properties of the existence and conjugacy of complements of a normal subgroup K of a finite group G are studied. The main results are as follows. (1) Suppose that K is abelian and every Sylow subgrop S of K has a complement in a Sylow subgroup of G which contains S. Then: (i) K has a complement in G; (ii) If G has a Hall π- subgroup H with π = π(K), and all complements of K in H are conjugate in H, then all complements of K in G are conjugate in G. (2) Suppose that K is solvable and K is a direct factor of S for each S/K∈ Syl(G/K).Then: (i) K has a complement in G;(ii) If G has a Hall π-subgroup H with π = π(K), then all complements of K inG are conjugate in G ff and only if all complements of K in H are conjugate in H.