Intersections of Three Nilpotent Subgroups in a Finite Group

Siberian Mathematical Journal - We complete the proof of the theorem that any nilpotent subgroups  $ A $ , $ B $ , and  $ C $ of a finite group  $ G $ satisfy...     

Criteria for $ \sigma $ -Solvability and Meta- $ \sigma $ -Nilpotency of a Finite Group

Siberian Mathematical Journal - We study some conditions for a group to be $ \sigma $ -solvable or meta- $ \sigma $ -nilpotent.     

有限群的$\Phi$-$\tau$-可补子群

假设$\tau$是一个子群算子, $H$是有限群$G$的一个$p$-子群. 令 $\bar{G}=G/H_{G}$且$\bar{H}=H/H_{G}$, 如果$\bar{G}$有一个次正规子群$\bar{T}$ 和一个包含于$\bar{H}$ 的$\tau$-子群$\bar{S}$满足$\bar{G}=\bar{H}\bar{T}$且$\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$, 就称$H$是$G$的一个$\Phi$-$\tau$- 可补子群. 文章通过讨论群$G$的准素数子群的$\Phi$-$\tau$-可补性给出了超循环嵌入和$p$-幂零性的一些新的特征.     

A New Characterization of Finite $ \sigma $ -Soluble $ P\sigma T $ -Groups

Siberian Mathematical Journal - We prove that  $ G $ is a finite $ \sigma $ -soluble group with transitive $ \sigma $ -permutability if and only if the following hold: (i) $ G $ possesses...     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

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.     

On Semi Cover-Avoiding Maximal Subgroups of Sylow Subgroups of Finite Groups

A subgroup H of a finite group G is called to have semi cover-avoiding property in G if there is a normal series of G such that H either covers or avoids every normal factor of the series. In this article we get some new results under the assumption that every maximal subgroup of Sylow subgroups of a suited subgroup of G has semi cover-avoiding property in G. We state our results in the broader context of formation theory.     

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.     

On a Finite Group Having a Normal Series Whose Factors Have Bicyclic Sylow Subgroups

We consider the structure of a finite group having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigate groups of odd order and A ₄-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.     

Subgroups of the Fan of Sylow Subgroups and the Supersolvability of a Finite Group

Mathematical Notes - The notion of the fan of a subgroup of a group, which was introduced in 1979 by Z. I. Borevich, is used to prove the supersolvability of finite groups. It is proved that a...     

On Intersections of $ pi $ -Hall Subgroups in Finite $ D_{pi} $ -Groups

Siberian Mathematical Journal - We give an example of a series of finite $ D_{pi} $ -groups where for each group $ G $ in the series and its $ pi $ -Hall subgroup $ H $ , the inequality $ Hcap...     

On the Average Number of Sylow Subgroups in Finite Groups

Czechoslovak Mathematical Journal - We prove that if the average number of Sylow subgroups of a finite group is less than $${{41} over 5}$$ and not equal to $${{29} over 4}$$ , then G is solvable...     

Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.     

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 ∈ ?.     

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.     

On s-Semipermutable Maximal and Minimal Subgroups of Sylow p-Subgroups of Finite Groups

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring R?_k k(X) in terms of the heights of simple modules with large endomorphism rings.     

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.     

有限群的X-s-置换子群

Let X be a nonempty subset of a group G.A subgroup H of G is said to be X-s-permutable in G if there exists an element x ∈ X such that HPx = PxH for every Sylow subgroup P of G.In this paper,some new results are given under the assumption that some suited subgroups of G are X-s-permutable in G.     

Finite Sylow Subgroups in Simple Locally Finite Groups of Lie Type

The main result of the paper is the followingTheorem. Let S = {r₀, r₁, ..., r_n} be a finite nonempty set of primes and let L be a Lie type of Chevalley groups. Then there exists a locally finite field F of characteristic r₀ such that Sylow r-subgroups of the simple group L(F) of type L over F are finite if and only if r ? S.