Levi子群L_α(n(α)=1)在Chevalley群G(Φ,F)中的扩群

设 Ｌ是复数域上单李代数，具有不可约根系 φ，固定基Ⅱ．设 Ｆ是一个特征不为２的域，且不是三元域，Ｇ（φ，Ｆ）是 Ｆ上φ型的 Ｃｈｅｖａｌｌｅｙ群．设 α∈Ⅱ，φα表示φ的一种类型子根系．当ｎ（α）＝１；且φ是Ｂｌ（ｌ ３），Ｄｌ（ｌ ４），Ｅ６，Ｅ７，或 Ｅ８之一时，本文决定了 Ｌｅｖｉ子群 Ｌα在 Ｇ（φ，Ｆ）中的所有扩群．     

Levi子群Lα(n(α)=1)在Chevalley群G(Ф,F)中的扩群

设L是复数域上单李代数,具有不可约根系Ф,固定基П.设F是一个特征不为2的域,且不是三元域,G(Ф,F)是F上Ф型的Chevalley群.设α∈П,Фα表示Ф的一种类型子根系.当n(α)=1,且Ф是Bl(l≥3),Dl(l ≥ 4),E6,E7,或E8之一时,本文决定了Levi子群Lα在G(Ф,F)中的所有扩群.     

Levi子群Lα（n（α）=1）在Chevalley群G（Φ，F）中的扩群

设L是复数域上单李代数，具有不可约根系Φ，固定基п。设F是一个特征不为2的域，且不是三元域，G（Φ，F）是F上Φ型的Chevalley群。设α∈п，Φα表示Φ的一种类型子根系。当n(α)＝1，且Φ是Bl(l≥3),Dl(l≥4),E6，E7或E8之一时，本文决定了Levi子群Lα在G（Φ，F）中的所有扩群。     

Chevalley群中Levi子群在抛物子群中的扩群

本文以根系及Ｗｅｙｌ群为基础,以换位子公式为工具,对任意域上任意类型Ｃｈｅｖａｌｌｅｙ群,针对极大抛物子群情形,确定了Ｌｅｖｉ子群在抛物子群中的所有扩群．     

D1和E6型Weyl群的扭子群的扩群

本决定了D1和E6型Weyl群扭子群的所有扩群，这为确定相应Chevalley群扭子群的所有扩群奠定了基础。     

关于A_(n-1)型CHEVALLEY扩群■的唯一性

本文讨论[1]中留下的A_(n-1)型Chevalley群G≌SL_n(K)的扩群G的唯一性问题,得到群G在同构意义下至多有1/2φ(n)个。     

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.     

有一个极大子群全正规的有限2群

给出了有一个极大子群全正规的有限2群的结构.     

Chevalley群的一类子群的研究

设G＝L（F）是特征不为2的域F上Chevalley群,型为B1（l≥4）,Cl（l≥3）,Dl（l≥5）,E6,E7,E8或F4之一．当L（F）型为B4或F4时还假设F=F2设Lα1是L（F）的一类Levy子群．本文决定Lα1的正规化子在L（F）中的极大性．     

次正规子群与有限群的超可解性

通过Sylow子群的极大子群和次正规性,利用极小阶反例的方法,得出群p-幂零性和超可解性的结论.本文的创新改进之处在于结合Sylow子群的极大子群和次正规性,研究p-幂零性和超可解性的相关结论.     

A New Type of Pronormal Subgroups in Chevalley Groups

Let L be a simple Lie algebra with irreducible root system φ having roots of different length,F be a field of characteristic different from 2,G=L(F) be a Chevalley group of type L over F.Denote by φ^1 the set of all long roots in φ.Set G^1=(zr(t);r∈φ^t,t∈F).It is a subgroup of G generated by all the long root subgroups.This paper determines the pronormality of G^1 in G when L is not of type G2.     

A NEW TYPE OF PRONORMAT SUBGROUPS IN CHEVALLEY GROUPS

51. IntroductionNotation in this paper follows [2,3]. Let L be a simple Lie algebra over an algebraicclosed field of characteristic 0,. its root system with a ford base A = {afl, or2,', al},and W the Weyl group of L. . (.--) denotes the set of all the positive (negative) roots.Each s E ac can be written piquely in the form 8 = Z aam with all the coefficients beingaamintegers. Moreover, either all ac 2 0 or all ac S 0. The sum Z ac of all these coefficielltsaamis called the height of 8, …     

Character Tables of Parabolic Subgroups of the Chevalley Groups of Type G 2

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G ₂(q), where q is a power of a prime p > 3.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

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.     

Universal Reflection Subgroups and Exponential Growth in Coxeter Groups

We investigate the imaginary cone in hyperbolic Coxeter systems in order to show that any Coxeter system contains universal reflection subgroups of arbitrarily large rank. Furthermore, in the hyperbolic case, the positive spans of the simple roots of the universal reflection subgroups are shown to approximate the imaginary cone (using an appropriate topology on the set of roots), answering a question due to Dyer [9 Dyer , M. Imaginary Cone and Reflection Subgroups of Coxeter Groups. Preprint: http://arXiv.org/abs/1210.5206  [Google Scholar]] in the special case of hyperbolic Coxeter systems. Finally, we discuss growth in Coxeter systems and utilize the previous results to extend the results of [16 Viswanath , S. ( 2008 ). On growth types of quotients of Coxeter groups by parabolic subgroups . Comm. Algebra 36 ( 2 ): 796 – 805 .[Taylor &; Francis Online] , [Google Scholar]] regarding exponential growth in parabolic quotients in Coxeter groups.     

有限域上F4和G2型Chevalley群的二元生成

本文证明了有限域上F4和G2型Chevalley群均可由两个元素生成.     

SL（2，q）的一个特征性质

本文证明了非单群系列ＳＬ（２，ｑ）（ｑ＝ｐ￣ｎ＞３）可以仅用其极大子群阶之集来刻划，从而得到了ＳＬ（２，ｑ）的一个特征性质．