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.     

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 Hall Normally Embedded Subgroups of Finite Groups

A subgroup H of a finite group G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H ^G . Groups which contain a Hall normally embedded subgroup of order d for every factor d of | G | are characterized. Such groups are supersolvable with a cyclic nilpotent residual of square-free order.     

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 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 ∈ ?.     

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.     

共轭置换、半正规与Sylow子群的极大子群

结合共轭置换与半正规的概念,研究了群G的Sylow子群的极大子群或共轭置换或半正规这一条件与群G的超可解性之间的关系.     

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.     

Locally Transitive Graphs Admitting a Group with Cyclic Sylow Subgroups

All graphs are finite simple undirected and of no isolated vertices in this paper. Using the theory of coset graphs and permutation groups, it is completed that a classification of locally transitive graphs admitting a non-Abelian group with cyclic Sylow subgroups. They are either the union of the family of arc-transitive graphs, or the union of the family of bipartite edge-transitive graphs.     

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.     

Finite Groups with Formation Subnormal Normalizers of Sylow Subgroups

Mathematical Notes - Let $$\mathfrak{F}$$ be a formation. Properties of the class $$\mathrm{w}^{*}\mathfrak{F}$$ of all groups $$G$$ for which $$\pi(G)\subseteq\pi(\mathfrak{F})$$ and the...     

关于具有给定Sylow子群正规化子的有限群Ⅱ

本文在有限可解群中解决了：任意ｍ－秩≤２的子群闭的局部群系具有性质：“如果群Ｇ的非单位Ｓｙｌｏｗ子群的正规化子属于，则群Ｇ也属于的一个充分必要条件．     

On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

l-群的极大下子群

在非正规值的条件下,给出l-群极大下子群的唯一表达形式,由此推广[1]中的一个结果.与此同时,建立了正规值l-群,特殊值l-群及有限值l-群之间等价的一个充要条件.