以$M_{11}$为基柱的旗传递点本原2-设计的完全分类

近年来,很多学者研究了以散在单群作为本原自同构群基柱的旗传递2-设计的一些分类工作.本文在此基础之上,给出了以散在单群$M_{11}$作为基柱的旗传递点本原2-设计的完全分类,得到了14个不同构的非平凡2-设计.     

交错群与旗传递点本原非对称2-（v，k，3）设计

受旗传递2-(v,k,3)对称设计和非对称2-(v,k,2)设计有关分类结果的启发,本论文继续研究旗传递非对称2-(v,k,3)设计.文章利用置换群的理论和组合设计的数量性质,借助计算机代数软件Gap和Magma,完全分类了自同构群G旗传递点本原,且基柱Soc(G)为交错群An(n≥5)的非对称2-(v,k,3)设计,证明了此类设计只能是唯一的2-(5,3,3)设计,且G=A_5或S_5.     

自同构群基柱为~3D_4(q)的2-(v,k,1)设计

2-(v,k,1)设计的存在性问题是组合设计理论中重要的问题,当这类设计具有一个有意义自同构群时,讨论其存在性是尤其令人感兴趣的.30年前,一个6人团队基本上完成了旗传递的2-(v,k,1)设计分类.此后,人们开始致力于研究区传递但非旗传递的2-(v,k,1)设计的分类课题.本文证明了自同构群基柱为~3D_4(q)的区传递及点本原非旗传递的2-(v,k,1)设计是不存在的.     

关于交换群上的Cayley有向图的正规性

Cayley有向图X=Cay(G,S)叫做正规的,如果G的右正则表示R(G)在X的全自同构群Aut(X)中正规,我们定出了交换群上的小度数的非正规的Cayley有向图, 并给出了一个猜想．应用这个结果,给出了pn(n≤2)个点上的度数不超过3的有向对称图的分类,这里p是一个奇素数．     

第二小阶数的双本原半对称图

如果一个正则图是边传递但不是点传递的,那么我们称它是半对称的.每一个半对称图X必定是两部分点数相等的二部图,并且它的自同构群Aut(X)在每一部分上是传递的.如果一个半对称图的自同构群在每一部分上作用是本原的,那么我们称它是双本原的.本文决定了第二小阶数的双本原半对称图.     

李型单群~2F_4(q~2)与2-(ν,κ,1)区组设计

本文证明了当2-(v,k,1)设计的自同构群G的基柱soc(G)=2F4(q2)时,Buekenhaut-Delandtsheer-Doyen猜想成立,即自同构群G的基柱为Ree群2F4(q2)的区本原2-(v,k,1)设计必为点本原的.     

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...     

容许二维线性群作用点本原的二弧传递图

一个图称为本原的如果它的自同构群作用在点集上是本原的.在这篇文章里,我们不但完全分类了容许一类二维线性群作用点本原的2-弧传递图,而且决定了他们的自同构群,并在同构意义下决定了它们的个数.     

李型单群2F4(q2)与2-((u),k,1)区组设计

本文证明了当2-((u),κ,1)设计的自同构群G的基柱soc(G)=2F4(q2)时,Buekenhaut-Delandtsheer-Doyen猜想成立,即自同构群G的基柱为Ree群2F4(q2)的区本原2-((u),k,1)设计必为点本原的.     

区传递的2-（v，k，1）设计与射影辛群PSpn（q）

分类自同构群为射影辛群PSpn(q)的区传递2-(v,k,1)设计,得到如下定理:设D为一个2-(v,k,1)设计,G≤Aut(D)是区传递,点本原但非旗传递的.若q为偶数且n≥14,则GPSpn(q).     

Construction of new symmetric designs with parameters (70,24,8)

This article presents a construction of all symmetric designs with parameters (70,24,8) on which the group E₈ F₂₁ operates so that the automorphism of order 7 operates fixed-point-free. A method based on enumeration of cosets in group is used for the construction.     

区传递的$2-(v, k, 1)$ 设计与单群$E_6(q)$

This article is a contribution to the study of block-transitive automorphism groups of 2-（v,k,1） block designs. Let D be a 2-（v,k,1） design admitting a block-transitive, pointprimitive but not flag-transitive automorphism group G. Let kr = （k,v-1） and q = pf for prime p. In this paper we prove that if G and D are as above and q （3（krk-kr ＋ 1）f）1/3, then G does not admit a simple group E6（q） as its socle.     

Quasi-symmetric 2-(28, 12, 11) designs with an automorphism of order 7

All quasi-symmetric 2-(28, 12, 11) designs with an automorphism of order 7 without fixed points or blocks are enumerated. Up to isomorphism, there are exactly 246 such designs. All but four of these designs are embeddable as derived designs in symmetric 2-(64, 28, 12) designs, producing in this way at least 8784 nonisomorphic symmetric 2-(64, 28, 12) designs. The remaining four 2-(28, 12, 11) designs are the first known examples of nonembeddable quasi-symmetric quasi-derived designs. These symmetric 2-(64, 28, 12) designs also produce at least 8784 nonisomorphic quasi-symmetric 2-(36, 16, 12) designs with intersection numbers 6 and 8, including the first known examples of quasi-symmetric 2-(36, 16, 12) designs with a trivial automorphism group. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 213–223, 1998     

Camina－Gagen定理的一个推广（Ⅱ）

设Ｇ是２－（ｖ，ｋ，１）设计Ｄ上的自同构群的一个子群，且是线－本原．如果（Ｖ，ｋ）＝ｋ／ｋ２，ｋ２≤１０．则Ｇ也是点－本原的。     

正交酉元列在有限von Neumann代数的迹自由积中的应用

令M_1为一个有限的von Neumann代数,τ_1为其上的一个忠实正规迹态.我们将证明,如果M_1中存在一列两两正交的酉元列{u_k:k∈N},则对任意具有忠实正规迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.作为推论可以得出,如果M_1有一个von Neumann子代数N不包含最小投影,则对任意具有忠实迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.     

Steinberg triality groups acting on 2 - (v, k, 1) designs

A 2 - (v,k,1) design D = (P, B) is a system consisting of a finite set P of v points and a collection B of k-subsets of P, called blocks, such that each 2-subset of P is contained in precisely one block. Let G be an automorphism group of a 2- (v,k,1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ≤ G ≤ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to 3D4(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.     

Sporadic groups and flag-transitive triplanes

Let D be a triplane, i.e., a 2-(v,k,3) symmetric design, and G be a subgroup of the full automorphism group of D. In this paper we prove that if G is flag-transitive point-primitive, then the socle of G cannot be a sporadic simple group.     

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 35)

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group PL(2, 3⁵).     

New symmetric 2-(176,50,14) designs

In this paper, we construct two new symmetric designs with parameters 2-$(176,50,14)$ as designs invariant under certain subgroups of the full automorphism group of the Higman design. One is self-dual and has the full automorphism group of order 11520 and the other one is not self-dual and has the full automorphism group of order 2520.