p-Fitting子群的CAP-嵌入子群

群G的子群H称为G的CAP-嵌入子群,如果对于|H|的每个素因子p,存在G的某个CAP-子群K,使得H的某个Sylow p-子群也是K的一个Sylowp-子群.本文通过假定G的p-Fitting子群F_p(G)的某个Sylow p-子群的每个极大子群是G的CAP-嵌入子群,得到一些新的结果.     

有限群的Fuzzy次正规子群与Fuzzy极大子群

本文研究了有限群的Ｆ次正规子群，得出了一个Ｆ子群是Ｆ次正规子群的充要条件，讨论了Ｆ次正规子群的一些重要性质。另外，本文还引入了有限群的Ｆ极大子群的概念，给出了Ｆ子群是Ｆ极大群的充要条件。最后，给出了三个定理，讨论了有限群Ｇ可解、超可解、幂零与Ｇ的Ｆ次正规子群、Ｆ极大子群之间的联系。     

Solitary Subgroups

We call a subgroup A of a finite group G a solitary subgroup of G if G does not contain another isomorphic copy of A. We call a normal subgroup A of a finite group G a normal solitary subgroup of G if G does not contain another normal isomorphic copy of A. The property of being (normal) solitary can be viewed as a strengthening of the requirement that A is normal in G. We derive various results on the existence of (normal) solitary subgroups.     

X-s-置换子群

设$X$是群$G$的一个非空子集.子群$H$在$G$中称为是$X$-$s$-置换的,如果对于$G$的每个Sylow子群$T$,存在一个元素$x\in X$,使得$HT^{x}=T^{x}H$.本文中,我们得到有关$X$-$s$-置换子群的一些结果,并利用它们刻画了一些有限群的结构.     

格序群的上下子群和Saturated子群的构造

本文对[1]中关于极大上下对的公开问题给出否定的回答并构造了例子说明[2]中命题2.1证明过程中的错误;研究了saturated子群的构造,得到了任意ι-群的正交子集生成的saturated子群以及投射ι—群的任一子集生成的saturated子群的具体形式.     

有限群的Fuzzy拟正规子群和Fuzzy次正规子群

讨论有限群的Fuzzy拟正规子群和Fuzzy次正规子群的一些性质。     

基于合意空间的模糊子群

利用合意空间(ConsensusSpace)理论给出了一个群的C模糊子群的定义.指出这种模糊子群实际上是基于t范(Tm范)的模糊子群.证明了Rosenfeld的模糊子群是C模糊子群,且每一个C模糊子群都与一类特殊的C模糊子群同构.从而为模糊子群提供了新的理论基础.     

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.     

关于SQ-补子群

假定Fitting子群F(G)或广义Fitting子群F^*(G)的某些子群在G中SQ-补来研究包含超可解群的饱和群系s,这里G∈s.一些已知结果被推广.     

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.     

Local Covering Subgroups in Finite Groups

A subgroup A of a finite group G is called a local covering subgroup of G if A~G=AB for all maximal G-invariant subgroup B of A~G=(A~g,g∈G).Let p be a prime and d be a positive integer.Assume that all subgroups of p~d,and all cyclic subgroups of order 4 when p~d=2 and a Sylow2-subgroup of G is nonabelian,of G are local covering subgroups.Then G is p-supersolvable whenever p~d=p or p~d≤(|G|_p)~(1/2) or p~d≤|O_(p'p)(G)|_p/p.     

Groups with Few Nonmodular Subgroups

Let G be a Tarski-free group such that the join of all nonmodular subgroups of G is a proper subgroup in G. It is proved that G contains a finite normal subgroup N such that the quotient group G/N has a modular subgroup lattice.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1419 – 1423, October, 2004.     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

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.     

Unique Node Extremal Subgroups in Chevalley Groups

Abstract

Taking G to be a Chevalley group of rank at least 3 and U to be the unipotent radical of a Borel subgroup B,an extremal subgroup A is an abelian normal subgroup of U which is not contained in the intersection of all the unipotent radicals of the rank 1 parabolic subgroups of G containing B. If there is an unique rank 1 parabolic subgroup P of G containing B with the property that A is not contained in the unipotent radical of P,then A is called a unique node extremal subgroup. In this paper we investigate the embedding of unique node extremal subgroups in U and prove that,apart from some specified cases,such a subgroup is contained in the unipotent radical of a certain maximal parabolic subgroup.     

On SS-Quasinormal Subgroups of Finite Groups

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.     

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.     

On Complemented Subgroups of Finite Groups

51. IntroductionIt is quite clear that the ekistence of complements for some families of subgroups of agroup gives a lot ofinfor~ion about its structure. FOr instance, Hall[6] proved that a groupG is supersoluble with elementary abelian Sylow subgroups if and only if G is complemellted,that is, every subgroup of G is comPlemeded in G. The same anchor also proved that agroup is soluble if and only if every Sylow subgroup is complemellted (see [3;I,3.5]). Morerecelltly, Arad and Wardll] pro…     

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.