New characterizations of finite supersoluble groups

Let A be a subgroup of a group G and X a nonempty subset of G.A is called an X- semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T_1 of T there exists an clement x∈X such that AT_i~x=T_i~xA.On the basis of this concept we obtain some new characterizations of finite supersoluble groups.     

关于有限群某些子群的X-半置换性

Let A be a subgroup of a group G and X a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we try to use the X-semipermutability of some subgroups to characterize the structure of finite groups.     

On Φ-τ-Supplement Subgroups of Finite Groups

Let τ be a subgroup functor and H a p-subgroup of a finite group G. Let G= G/H_G and H= H/H_G. We say that H is Φ-τ-supplement in G if G has a subnormal subgroup T and a τ-subgroup S contained in H such that G=H T and H∩T≤SΦ(H). In this paper,some new characterizations of hypercyclically embedability and p-nilpotency of a finite group are obtained based on the assumption that some primary subgroups are Φ-τ-supplement in G.     

关于有限群的$CAP$-嵌入子群

A subgroup H of a finite group G is said to be CAP-embedded subgroup of G if, for each prime p dividing the order of H, there exists a CAP-subgroup K of G such that a Sylow p-subgroup of H is also a Sylow p-subgroup of K. In this paper some new results are obtained based on the assumption that some subgroups of prime power order have the CAP-embedded property in the group.     

Finite p-Groups with a Class of Complemented Normal Subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

On weakly S-embedded subgroups of finite groups

Let H be a subgroup of a group G.Then H is said to be S-quasinormal in G if HP = P H 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 ∩ T≤H SE,where H SE denotes the subgroup of H generated by all those subgroups of ...     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

Cuccia定理的一个推广

For a finite group G,let S(G)be the set of minimal subgroups of odd order of G which are complemented in G.It is proved that if every minimal subgroup X of odd order of G which does not belong to S(G),C_G(X)is either subnormal or abnormal in G.Then G solvable.     

On Certain Distributive Lattices of Subgroups of Finite Soluble Groups

In this paper, we prove the following result. Let ξ be a saturated formation and ∑ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that ∑ reduces into each element of X. Consider in G the following three subgroups： the ξ-normalizer D of G associated with ∑; the X-prefrattini subgroup W = W（G, X） of G; and a hypercentrally embedded subgroup T of G. Then the lattice ζ（T, W, D） generated by T, D and W is a distributive lattice of pairwise permutable subgroups of G with the cover and avoidance property. This result remains true for the lattice ,ζ（V, W, D）, where V is a subgroup of G whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups of G such that ∑ reduces into V.     

有限群的弱S-拟正规嵌入子群

群G的子群H称为在G中S-拟正规嵌入的,如果对于任意的素数p||H|,H的Sylow p-子群也是G的某个S-拟正规子群的Sylow p-子群.称群G的子群H在G中弱S-拟正规嵌入,如果存在群G的正规子群T,使得HT■G且H∩T在G中是S-拟正规嵌入的.研究了弱S-拟正规嵌入子群的性质,给出了某些群类的新的特征,并推广了一些已知的结论.     

具有若干$c_p$-可补子群的有限群

设$H$为有限群$G$的子群且$p$为整除群$G$的阶的素因子. 我们称$H$在$G$中为$c_p$-可补的,如果$G$中存在$H$的包含$H_G$的补子群$T$使得$H\cap T/H_G$为$p''$-群, 其中$H_G$为$H$在$G$中的核. 群$G$ 称$CS_p$-群, 如果$G$的所有$p$-子群都在$G$中$c_p$-可补. 本文,我们刻画具有若干$c_p$-可补$p$-子群的有限群的$p$-可解性和$p$-超可解性. 此外,我们给出$p$-可解群为$CS_p$-群的若干等价条件. 最后, 我们给出两个$CS_p$-群的直积为$CS_p$-群的判别准则. 我们的结果推广了近期的若干结论.     

有限群的$p$-覆盖远离子群及$S$-拟正规嵌入子群

Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p-subgroup of G. If d is the smallest generator number of P, then there exist maximal subgroups P1, P2,..., Pd of P, denoted by Md(P) = {P1,...,Pd}, such that di=1 Pi = Φ(P), the Frattini subgroup of P. In this paper, we will show that if each member of some fixed Md(P) is either p-cover-avoid or S-quasinormally embedded in G, then G is p-nilpotent. As applications, some further results are obtained.     

p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.     

2-极大子群的$F_s$-拟正规性

$F$是一个群系. $G$的子群$H$在$G$中称为$F_s$-拟正规的,如果存在$G$的正规子群$T$,使得$HT$在$G$中是$s$-置换的并且$(H\cap T)H_G/H_G$包含在$G/H_G$的$F$超中心$Z^F_\infty(G/H_G)$中.本文利用$F_s$-拟正规子群研究了有限群的结构.获得了某些新的结果.     

关于$\mathcal{F}$-S-可补子群

设$\mathcal{F}$是一个群类. 群$G$的子群$H$称为在$G$中$\mathcal{F}$-S-可补的，如果存在$G$的一个子群$K$,使得$G=HK$且$K/K\cap{H_G}\in\mathcal{F}$, 其中$H_G=\bigcap_{g\in G}H^g$是包含在$H$中的$G$的最大正规子群．本文利用子群的$\mathcal{F}$-S-可补性, 给出了有限群的可解性, 超可解性和幂零性的一些新的刻画. 应用这些结果, 我们可以得到一系列推论, 其中包括有关已知的著名结果.     

闭2-极大子群指数为素数幂的有限群

设G为有限群, 如果G的每个非2-闭极大子群的指数均为素数幂, 那么 G-S(G)≌PSL2(7)或1, 其中S(G)为G的最大可解正规子群.     

具有唯一非平凡正规子群的有限群的Coleman自同构

Let G be a finite group with a unique nontrivial normal subgroup. It is shown that every Coleman automorphism of G is an inner automorphism.     

The Influence of Primitive Subgroups on the Structure of Finite Groups

A subgroup $H$ of a group $G$ is said to be primitive if it is a proper subgroup of the intersection of all subgroups of $G$ containing $H$ as its proper subgroup. The purpose of this note is to go further into the influence of primitive subgroups on the structure of finite groups. Some new results are obtained.     

The influence of minimal subgroups on the structure of a finite group

We study the detailed structure of a finite group under the assumption that all minimal subgroups of the generalized Fitting subgroup of some normal subgroup of are well-suited in .