作用在有限线性空间上基柱为3D4(q) 的几乎单群

旗传递线性空间的分类完成以后, 人们开始关注线传递线性空间. 线传递线性空间可以分为非点本原和点本原两种情形. 根据 Delandtsheer-Doyen 理论, 非点本原线传递分类比较容易解决. 而点本原的情形, 根据 O''Nan-Scott 理论和 Camina 的一些前期工作, 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 T 是非交换单群, T≤ G≤ Aut(T) 且 G 线传递作用在有限线性空间上的情形. 并获得了一些有用的引理. 特别地, 证明了当 T 同构于 ³D₄(q) 时, T 是线传递的, 这里 q 是素数 p 的方幂.     

Lie型单群3D4(q)和2-(v,k,1)设计

设D是一个2-(v,k,1)设计,G是D的自同构群.Delandtsheer证明了如果G是区本原的,且D不是射影平面,则G是几乎单群,即存在一个非交换单群T,使得T≤G≤Aut(T).本文证明了T不同构于单群3D4(q),这是区本原设计分类工作的一个不可缺少的组成部分.     

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。     

区传递的2-(ν,κ,1)设计与李型单群E8(q)

分类自同构群的基柱为李型单群E8(q)的区传递2-(ν,κ,1)设计,得到如下定理:设D为一个2-(ν,κ,1)设计,G≤Aut(D)是区传递、点本原但非旗传递的.若q＞24√(krk-kr 1)f(这里kr=(k,v-1),q=pf,p是素数,f是正整数),则Soc(G)(≠)E8(q).     

若干单群与射影平面直射群

设G是有限群,H(?)G。如果H≌~2B_2(q)或H≌~2G_2(q)或H≌PSU(3,q),则G不与任何射影平面的点传递直射群同均。本文对以下问题给出了一般方法:证明以某些几乎单群为点传递自同构群的线性空间不是射影平面。     

区传递的2-(v,9,1)设计的分类

讨论区传递的2-(v,k,1)设计的分类问题.特别地,讨论自同构群的基柱为典型单群的区传递,点本原但非旗传递的2-(v,9,1)设计.设D为一个2-(v,9,1)设计,若G≤Aut(D)是区传递,点本原但非旗传递的,则G的基柱Soc(G)不是有限域GF(q)上的典型单群.结合Camina,Praeger,刘伟俊,李慧陵...     

作用在有限线性空间上基柱为2F4(q)的几乎单群

本文研究了基柱同构于2F4(q)的线传递的几乎单群.利用了有限线性空间上线传递的自同构群的经典结论,以及2F4(q)的阶和极大子群等性质,获得了点稳定子群和线稳定子群的分类.     

有限单群的局部本原Cayley图

设G是图Γ的全自同构群的一个子群,Γ称为是G-局部本原的,如果顶点α的点稳定子群G_α在α的邻域Γ(α)上作用本原.对于非交换单群L和它的一个Cayley子集S,假设L(G≤Aut(L),且相应的Cayley图Γ=Cay(L,S)是G-局部本原的.证明了这时L必为一个Lie型单群,且或者Γ的度数为|Out(L)|的奇素数因子,或者L=PΩ⁺₈(q)而Γ的度数为4.还证明了在这两种情形下Γ的全自同构群都是以L为基座的几乎单群.     

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.     

Lie型单群3D4(q)和2-( v,k, 1)设计

设D是一个2-(v, k, 1)设计, G是D的自同构群. Delandtsheer证明了如果G是区本原的, 且D不是射影平面, 则G是几乎单群, 即存在一个非交换单群T , 使得T≤G≤Aut(T). 本文证明了T不同构于单群³D₄(q), 这是区本原设计分类工作的一个不可缺少的组成部分.     

Almost simple groups with socle 3D4(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.     

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.     

类对称函数的Schur凸性和不等式

对x = (x₁, x₂,···, x_n) ∈ (0,1)ⁿ 和 r ∈ {1, 2,···, n} 定义对称函数 F_n(x, r) = F_n(x₁, x₂,···, x_n; r) =∏_{1≤i1∑j=1^r(1+xi3/1- xi3)^1/r, 其中i1, i2, ···, ir 是整数. 该文证明了Fn(x, r) 是(0,1)ⁿ 上的Schur凸、Schur乘性凸和Schur调和凸函数. 作为应用,利用控制理论建立了若干不等式.}     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

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.     

Quasirecognizability by the Set of Element Orders for Groups 3 D 4(q), for q Even

It is proved that if G is a finite group with an element order set as in the simple group ³D₄(q), where q is even, then the commutant of G/F(G) is isomorphic to ³D₄(q) and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 3–19, January–February, 2006.     

Analysis on discrete cocompact subgroups of the generic filiform Lie groups

Summary Let F_n, n≧ 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each F_n, the infinite dimensional simple quotients of the group C*-algebra of (the most obvious) one of its discrete cocompact subgroups D_n. For D_n, the most attractive concrete faithful representations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H₃ ⊆G₃ on L²(T) that result from the irrational rotation flows on T; the representations of D_n generate infinite-dimensional simple quotients A_n,θ of the group C*-algebra C*(D_n). For n>1, there are other infinite-dimensional simple quotients of C*(D_n) arising from non-faithful representations of D_n. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group C*-algebras of the lower dimensional tori.     

Separation properties for Sturm-Liouville operators

Let q (x) be a positive function given on the interval I of the real axis; let P be the minimal operator generated in L₂(0, +) by the differential expression P[·]=-d²/dx²+q(x); let Q be the operator of multiplication by the function q(x). If D_p* Dq, then P [·] is said to be separated. In this note the separation of P[·] is proved for some growth regularity conditions on the fonction q (x), without assuming anything on its smoothness. One proves that if D_p* D_S, where S is the multiplication operator by the function s(x), satisfying some growth regularity condition, then D_Q D_S.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 349–359, September, 1973.     

Parabolic permutation representations of the groups2 F 4(q) and3 D 4(q 3)

In the paper, the ranks, degrees, subdegrees, and double centralizers of permutation representations of the bounded groups² F ₄(q) and³ D ₄(q ³) with respect to parabolic maximal subgroups of nonminimal index are found. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 69–76, January, 2000.