作用在有限线性空间上基柱为F_4(q)的几乎单群

线传递线性空间可以分为非点本原和点本原两种情形,而点本原的情况又可以分成基柱为初等交换群或非交换单群两种情形.本文考虑后一种情形,即T是非交换单群,T≤G≤Aut(T)且G线传递,点本原作用在有限线性空间上的情形.证明了当T同构于F_4(q)时,若T_L不是~2F_4(q),B_4(q),D_4(q).S_3,~3D_4(q).3,F_4(q~(1/2))和T的抛物子群的子群时,T也是线传递的,这里q是素数p的方幂.     

旗传递5-(v,k,2)设计的分类

本文研究了5-(v,k,2)设计的分类问题.利用典型群PSL(2,q)的子群作用于投影线的轨道定理,证明了旗传递5-(v,k,2)设计的自同构群的基柱不能与PSL(2,3n)同构.从而证明了不存在旗传递的5-(v,k,2)设计.     

某些Lie单群的传递性

设D为有限线性空间，且T G Aut（T），其中T是非交换单群，并且同构于^2B2（g），Cn（g）（n≥3），^3D4（g），E7（q），E8（q），F4（q），^2F4（q），G2（q），^2G2（q）。假设D不是射影平面，G线传递作用在D上，则T点传递。     

F-S-可补的子群对有限群结构的影响

设F是一个群系.群G的一个子群H在G中F-S-可补,如果存在G的子群K,使得G=HK且K/K∩HG∈F,其中HG表示G包含在H中的最大的正规子群.本文利用群系理论研究子群的F-S-可补性对有限群结构的影响,得到如下结论:设F是子群闭的局部群系,G是有限群且GF是可解的.则G∈F的充要条件是下列条件之一:(1)G存在正规子群N使得G/N∈F且N的极小子群及4阶循环子群(p=2)均在G中F-S-可补.(2)G存在正规子群N使得G/N∈F,N的4阶循环子群在G中有F-S-补且N的极小子群皆包含在Z∞F(G)中.应用这些结论,可以得到一些推论,其中包括已知的相关结果.     

Abe-Iiyori猜想和Ree群2F4(q)(Ⅱ)

就一类单群2F4(q)和2F4(2)'证明了Abe-Iiyori猜想.     

森谱的界

设λ_k(F)是树或者森的第k大特征值,[x]是不超过x的最大整数,q是F的边独立数.本文证明了:对于1≤k≤[(q 1)/2]有λ_k(F)≥1,并且这个下界是最好可能的;对于1≤i≤[q/2],若q为偶数,则有λ[(q 1)/2] i(F)≥2cos((2iπ)/(4i 1)),若q为奇数,则有λ_([(q 1)/2] i)(F)≥2cos(((2i 1)π)/(4i 3)),     

一道高考题的多种解法

题目 (2000年全国高考题 ):过抛物线y=ax2 (a>0)的焦点F作一直线交抛物线于P、Q两点,若线段PF、FQ的长分别是p、q,则1p+1q等于(　　)(A) 2a　　 (B)12a　　 (C) 4a　　 (D)4a思路 1　抓住“过焦点F作一直线交抛物线于P、Q两点”这一条件,利用特殊位置,可获得简捷解法.　　解法 1　由y=ax2 得x2 =1ay,于是抛物线的焦点为F 0,14a,如图,取过点F且平行于X轴的直线与抛物线交于P、Q两点,显然PF=FQ,即p=q,设Qx,14a,将其代入抛物线方程易求得x=12a. 　∴p=q=12a,即1p+1q=4a,故应选C(　　).思路 2　题目给定的已知条件“线段PF,PQ的…     

有限可解(q)群与(s—q)群的一个注记

文[l].[2]分别研究了每个次正规子群为拟正规的有限群(即(q)群)以及每个次正规子群为s—q拟正规的有限群(即(s—q)群).本文利用广幂零群的概念对(q)群与(s—q)群给出了一个新的刻划,并得到内(s—q)群的完全分类。     

Suzuki群Sz(q)和线性空间的自同构群

本文研究了线性空间的几乎单的线传递自同构群.利用有限线性空间上线传递自同构群的经典结论,以及Suzuki群Sz(q)的性质,获得了线性空间上线传递且点本原的自同构群的基柱不是Sz(q)的结果,推广了关于线传递性空间的已有结果.     

二维射影线性群与区传递4-(v,6,λ)设计

设D=(X,B)是一个4-(v,6,λ)设计,GAut(D)区传递地作用在D上且X=GF(q)∪{∞},这里GF(q)是q元有限域.如果G=PSL(2,q),则存在4-(12,6,4)设计;如果G=PGL(2,q),则存在4-(12,6,8),4-(18,6,24)和4-(33,6,12)设计.     

Almost simple groups with socle ~3D_4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O'Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T≤G≤Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to 3D4(q), then T is line-transitive, where q is a power of the prime p.     

Finite line-transitive linear spaces: Theory and search strategies

The paper summarises existing theory and classifications for finite line-transitive linear spaces, develops the theory further, and organises it in a way that enables its effective application. The starting point is a theorem of Camina and the fifth author that identifies three kinds of line-transitive automorphism groups of linear spaces. In two of these cases the group may be imprimitive on points, that is, the group leaves invariant a nontrivial partition of the point set. In the first of these cases the group is almost simple with point-transitive simple socle, and may or may not be point-primitive, while in the second case the group has a non-trivial point-intransitive normal subgroup and hence is definitely point-imprimitive. The theory presented here focuses on point-imprimitive groups. As a non-trivial application a classification is given of the point-imprimitive, line-transitive groups, and the corresponding linear spaces, for which the greatest common divisor gcd（k, v - 1） ≤ 8, where v is the number of points, and k is the line size. Motivation for this classification comes from a result of Weidong Fang and Huffing Li in 1993, that there are only finitely many non-trivial point-imprimitive, linetransitive linear spaces for a given value of gcd（k, v - 1）. The classification strengthens the classification by Camina and Mischke under the much stronger restriction k ≤ 8： no additional examples arise. The paper provides the backbone for future computer-based classifications of point-imprimitive, line- transitive linear spaces with small parameters. Several suggestions for further investigations are made.     

A conjecture of Abe-Iiyori and the Ree groups ~2F_4(q),Ⅱ

In this paper the author has solved a problem of Abe and liyori for the finite simple groups 2F4(q) and 2F4(2)'.     

A conjecture of Abe-Iiyori and the Ree groups 2F4(q),Ⅱ

In this paper the author has solved a problem of Abe and liyori for the finite simple groups 2F4(q) and 2F4(2)'.     

Zum Stufenaufbau der Inzidenzstrukturen mit Ähnlichkeitsrelation

Leißner [2] proved that the class of all incidence structures with similarity-relation coinzides with, the class of the algebraically defined geometries [F,T], where F denotes a neardomain over a subdomain T.In this paper we characterize those geometries, where F is a near-resp. (skew-) field by additional similarity axioms. At first we show that a subdomain T of a neardomain F is itself a neardomain iff–1T and characterize this fact geometrically. As a consequence every subdomain of a near-resp.(skew-) field has to be a near-resp. (skew-) field too. In §4 we get as a corollary that projective planes admit no sharply twice transitive groups of collineations.     

On Primitive Representations of Finite Alternating and Symmetric Groups with a 2-Transitive Subconstituent

In this paper we analyse primitive permutation representations of finite alternating and symmetric groups which have a 2-transitive subconstituent. We show that either the representation belongs to an explicit list of known examples, or the point stabiliser is a known almost-simple 2-transitive group and acts primitively in the natural representation of the associated alternating or symmetric group.     

Suzuki-Ree群的自同构群的阶分量刻画

在《数学学报》2013年第56卷第4期中,"Suzuki-Ree群的自同构群的一个新刻画"一文证明了Aut(~2F_4(q)),q=2~f和Aut(~2G_2(q)),q=3~f,可由其阶分量刻画,其中f=3~s,s为正整数.本文证明了Aut(~2B_2(q)),q=2~f和Aut(2G2(q)),q=3~f,也可由其阶分量刻画,其中f为奇素数.结合二者得到结论:Suzuki-Ree单群的所有的素图不连通的自同构群皆可由其阶分量刻画.     

Almost simple groups with socle 3 D 4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.