On c-normality of certain subgroups of prime power order of finite groups

In this paper, we study the structure of the finite group G given that certain subgroups of prime power order are well-situated, which means that they are normally complemented modulo their normal core.Received: 14 October 2004; revised: 12 January 2005     

某些C-正规子群对有限群结构的影响

利用某些子群的C正规性,得到有限群成为可解的一系列充分条件     

On finite groups with some subgroups of prime indices

We establish the solvability of each finite group whose every proper nonmaximal subgroup lies in some subgroup of prime index.     

F-S-可补的子群对有限群结构的影响

设F是一个群系.群G的一个子群H在G中F-S-可补,如果存在G的子群K,使得G=HK且K/K∩HG∈F,其中HG表示G包含在H中的最大的正规子群.本文利用群系理论研究子群的F-S-可补性对有限群结构的影响,得到如下结论:设F是子群闭的局部群系,G是有限群且GF是可解的.则G∈F的充要条件是下列条件之一:(1)G存在正规子群N使得G/N∈F且N的极小子群及4阶循环子群(p=2)均在G中F-S-可补.(2)G存在正规子群N使得G/N∈F,N的4阶循环子群在G中有F-S-补且N的极小子群皆包含在Z∞F(G)中.应用这些结论,可以得到一些推论,其中包括已知的相关结果.     

The influence of minimal subgroups on the structure of finite groups

We are interested to study the structure of a finite group G under the assumption that all minimal subgroups of G are well-situated in G.     

The influence of weakly-supplemented subgroups on the structure of finite groups

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In the paper it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of P∩G′ is weakly-supplemented in N _G (P), where P is a Sylow p-subgroup of G. As applications, some interesting results with weakly-supplemented minimal subgroups of P∩G′ are obtained.     

The influence of minimal subgroups of focal subgroups on the structure of finite groups

A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in N_G(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained.     

S-quasinormality of finite groups

Let d be the smallest generator number of a finite p-group P, and let be a set of maximal subgroups of P such that . In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SS-quasinormal subgroups in , for each prime p, and Sylow p-subgroups P of G is studied.     

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be 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 investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.     

A note on character kernels in finite groups of prime power order

In this note, we classify the finite groups of prime power order for which all nonlinear irreducible character kernels constitute a chain with respect to inclusion. Received: 6 April 2007     

On some primary subgroups and the structure of finite groups

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$ . $H$ is said to be an $s$ -quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$ , a Sylow $p$ -subgroup of $H$ is also a Sylow $p$ -subgroup of some $S$ -quasinormal subgroup of $G$ ; $H$ is said to be $c$ -normal in $G$ if $G$ has a normal subgroup $T$ such that $G=HT$ and $H\cap T\le H_{G}$ , where $H_{G}$ is the normal core of $H$ in $G$ . We fix in every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$ satisfying $1<|D|<|P|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either $s$ -quasinormally embedded or $c$ -normal in $G$ . Some recent results are generalized and unified.     

The influence of s-conditionally permutable subgroups on finite groups

A subgroup H of a group G is called s-conditionally permutable in G if for every Sylow subgroup T of G there exists an element x ∈ G such that HTx = TxH. Using the concept of s-conditionally permutable subgroups, some new characterizations of finite groups are obtained and several interesting results are generalized.     

Maximal cyclic subgroups and prime divisors in finite groups

Let G be a finite group with no chief factor simple of Lie type E ₈(q) and C a cyclic subgroup of largest order in G. It is shown that at most two primes in the open interval ([|C|/2], |C|) divide |G|.Received: 14 January 2005     

三个或多个素变数的线性方程

设G是有限群,p是|G|的一个素因子,P是G的一个Sylow p-子群.若下列条件之一满足,则G是p-幂零:(1)P的极大子群均在G中S-半正规且(|G|,p-1)=1;(2)P的二次极大子群均在G中S-半正规且(|G|,p2-1)=1.