关于有限群的弱S-可补嵌入子群(英文)

假定H是有限群G的一个子群.如果对于|H|的每个素因子p,H的一个Sylow p-子群也是G的某个s-可换子群的Sylow p-子群,则称H为G的s-可换嵌入子群;如果存在G的子群T使得G=HT并且H∩T≤HG,其中HG为群G含于H的最大的正规子群,则称H为G的c-可补子群;如果存在G的子群T使得G=HT并且H∩T≤Hse,其中Hse为群G含于H的一个s-可换嵌入子群,则称H为G的弱s-可补嵌入子群.本文研究弱s-可补嵌入子群对有限群结构的影响.某些新的结论被进一步推广.     

有限群的弱正规子群与p-幂零性

研究了p2阶子群以及一般的pk阶子群为弱正规子群时有限群G的结构.给出了有限群为p-幂零群以及超可解群的一些条件.     

具有p-幂零s-补子群的有限群（英文）

有限群G的子群H叫做F-s-补子群,若存在G的一个子群K使得G=HK且K/(K∩H_G)∈F,其中F是一个群类.本论文利用p-幂零s-补子群得到了关于有限群为p-幂零群的一些新成果.     

关于F-S-可补子群

设F是一个群类．群G的子群H称为在G中F-S-可补的,如果存在G的一个子群K,使得G=HK且K/K∩HG∈F,其中HG=∩g∈GHg是包含在H中的G的最大正规子群．本文利用子群的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．     

有限群的Sylow-子群的弱s-可补极大子群(英文)

假设G是一个有限群,H是G的一个子群.称H在G是s-置换的,若对G的任意的Sylow-子群Gp,有HG_p=G_pH:称H在G是弱s-可补的,若存在G的子群T使得G=HT且H∩T≤H_(sG),其中H_(sG)是所有包含在H中的G的s-置换子群生成的子群.本文给出了下列定理:设F是一个包含超可解群系u的饱和群系,有限群G有一个正规子群H使得G/H∈F.若F~*(H)的每个Sylow子群的所有极大子群在G中是弱s-可补的,其中F~*(H)是H的广义Fitting子群,则G∈F.它是J.Algebra,2007,315:192-209一文中的Skiba公开问题在极大子群情形下的肯定回答.     

弱s-拟正规子群对有限群的p-幂零性的影响

群G的子群H称为G的弱s-拟正规子群,若G有次正规子群T,使得G=HT且H ∩T≤HsG,其中HsG是包含在H中的G的最大的s-拟正规子群.本文利用Sylow p-子群的极大子群的弱s-拟正规性得到有限群为p-幂零群的一些充分条件,并给出Schur-Zassenhaus定理的一种推广.     

弱s-置换性传递的有限群

群G被称为弱s-置换性传递的群,对于它的子群H和K,若H在K中弱s-置换,K在G中弱s-置换,则H在G中弱s-置换.本文给出弱s-置换性、弱s-补性传递的可解群的结构以及每一子群在G中弱s-置换、弱s-补的群的结构.     

具有幂零局部子群的有限群

一个有限非幂零群G称为PN-群,如果NG(P)是幂零的,对于每个素数p和每个满足P(∈)Z∞(G)的非正规子群p-子群P.本文将给出可解PN-群的结构和一些特征定理.     

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.     

具有幂零局部子群的有限群

一个有限非幂零群G称为PN-群,如果NC(P)是幂零的,对于每个素数p和每个满足PZ∞(G)的非正规子群p-子群P.本文将给出可解PN-群的结构和一些特征定理.     

The Influence of Weakly s-Supplemented Subgroups on the Structure of Finite Groups

Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _{s G} , where H _{s G} is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized.     

On Weakly s-permutably Embedded Subgroups of Finite Groups

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutably 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-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H _se of G contained in H such that G = HT and H ∩ T ≤ H _se . We investigate the influence of weakly s-permutably embedded subgroups on the structure of finite groups. Some recent results are generalized.     

A Criterion of p-Nilpotency of Finite Groups

Let G be a finite group and p a prime. In this article, we give a criterion of p-nilpotency of G by means of the normalizer and a generalization of normality of subgroups.     

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.     

On SQ-Supplemented Subgroups of Finite Groups

Let G be a finite group and H ≤ G. The subgroup H is called: S-permutable in G if HP = PH for all Sylow subgroups P of G; S-permutably embedded in G if each Sylow subgroup of H is also a Sylow subgroup of some S-permutable subgroup of G.

Let H be a subgroup of a group G. Then we say that H is SQ-supplemented in G if G has a subgroup T and an S-permutably embedded subgroup C ≤ H such that HT = G and T ∩ H ≤ C.

We study the structure of G under the assumption that some subgroups of G are SQ-supplemented in G. Some known results are generalized.     

A Characteristic Property of Finite p -Nilpotent Groups

Abstract

An element x of a group G is called Q-central if there exists a central chief factor H/K of G such that x ∈ H\K. It is proved that a finite group G is p-nilpotent if and only if every element in G _p  \Φ(G _p ) is Q-central. We will adapt Doerk and Hawkes [Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin–New York: Walter de Gruyter, pp. 892] for notations and basic results.     

Finite Groups with Some S-quasinormally Embedded Subgroups

Suppose that G is a finite group and H is a subgroup of G. H is said to be S-quasinormal in G if it permutes with every Sylow 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. We investigate the influence of S-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.     

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.