On F-Abnormal Maximal Subgroups of Finite Groups

For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.     

非幂零极大子群共轭类类数给定的有限群

证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画.     

二次极大子群中2阶及4阶循环子群拟正规的有限群

本文讨论２阶及４阶循环子群对群结构的影响．主要结果是下述定理：如果有限群Ｇ满足标题的条件，那么下列情形之一成立：（１）Ｇ有正规Ｓｙｌｏｗ ２－子群；（２） Ｇ为 ２－幂零；（３） Ｇ ≌ Ｓ４；（４） Ｇ＝ＰＱ，其中 Ｐ为阶 ２４广义四元数群， Ｑ为 ３阶循环群；（５） Ｇ ≌ Ａ５或 ＳＬ（２，５）．     

Maximal and Sylow Subgroups of Solvable Finite Groups

The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.     

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F*1v.     

n-极大子群为C-正规的有限群

首先,利用Hall子群的C－正规性,得到了有限群成为。可解群的一个充分条付,推广了Schur-Zassenhaus定理;其次,考查了 n－极大子群或其 2阶及 4阶循环子群为 C-正规对有限群结构的影响．     

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.     

Finite Groups with Subnormal Schmidt Subgroups

We study finite groups some of whose Schmidt subgroups (the minimal nonnilpotent subgroups) are subnormal.     

子群次正规性对有限群可解性的影响

考查了次正规子群对有限群结构的影响,得到有限群可解的若干充分条件和超可解的一个充分条件.     

有限生成的无限可解群的多余子群

本文得到了有限生成的无限可解群的多余子群的一些结果，它们是有限群的某些相应结果的推广。     

欧氏环上特殊正交群的一类极大子群

定出欧氏环上特殊正交群的一类极大子群,得到如下结果:设R是带有欧氏映射σ的特征不为2的欧氏环且不是域,SO(2m,R)为R上的特殊正交群,R*=R\{0},l=min{σ(x)|x∈R*\U(R)},任取a∈R*\U(R)使σ(a)=l,记a在R中生成的主理想为M.那么当m≥3时,AB CD∈SO(2m,R)|B∈Mm×m是SO(2m,R)的一个极大子群.     

On c -Normal Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H _G  = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are:

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ?.

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ?.     

有限群的c-半置换子群

研究了有限群的结构问题,利用子群c-半置换和完全c-半置换的定义和性质,通过对有限群sylow子群的2-极大子群的研究,获得了有限群幂零、p-幂零的充分条件和另外两个决定群结构的充要条件.     

最大Abel商群为局部循环群的可解群

从自同构群的角度出发,给出了一些具有有限性条件的、最大Abel商群为局部循环群的可解群的结构．     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

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.     

On $\mathfrak{F}_\tau$-$s$-Supplemented Subgroups of Finite Groups

Let $\mathfrak{F}$ be a non-empty formation of groups, $\tau$ a subgroup functor and $H$ a $p$-subgroup of a finite group $G.$ Let $\overline{G}=G/H_G$ and $\overline{H} =H/H_G.$ We say that $H$ is $\mathfrak{F}_\tau$-$s$-supplemented in $G$ if for some subgroup $\overline{T}$ and some $\tau$-subgroup $\overline{S}$ of $\overline{G}$ contained in $\overline{H},$ $\overline{H}\overline{T}$ is subnormal in $\overline{G}$ and $\overline{H} ∩ \overline{T} ≤ \overline{S}Z_{\mathfrak{F}}(\overline{G}).$ In this paper, we investigate the influence of $\mathfrak{F}_\tau$-$s$-supplemented subgroups on the structure of finite groups. Some new characterizations about solubility of finite groups are obtained.     

Finite Groups with S-permutable n-maximal Subgroups

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n?1(G) of order 4 are S-permutable in G.     

关于一类极大子群的θ-偶

对于有限群G的极大子群M，令β（G：M）表示整除│G：M│的素因子个数，β（G）表示所有β（G；M）中的最大数．令μ（G）为使得β（G：M）=β（G）的极大子群的集合．通过对这一类极大子群的θ-偶赋予一定条件，得到了判断群G可解、超可解的新结果．