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.     

Finite classical groups and flag-transitive triplanes

Let D be a finite nontrivial triplane, i.e. a 2-(v,k,3) symmetric design, with a flag-transitive, point-primitive automorphism group G. If G is almost simple, with the simple socle X of G being a classical group, then D is either the unique (11, 6, 3)-triplane, with G=PSL₂(11) and G_α=A₅, or the unique (45, 12, 3)-triplane, with G=X:2=PSp₄(3):2≅PSU₄(2):2 and , where α is a point of D.     

典型群PSU(3,q~2)与2-(v,k,1)设计

讨论自同构群是酉群 PSU(3 ,q2 ) (q=2 l)的区 -本原的 2 -(v,k,1 )设计 .首先证明了它必是点 -本原的 ,然后确定了这种类型的设计 ,即它只能为 2 -(q3+1 ,q+1 ,1 )设计     

Symmetric designs and four dimensional projective special unitary groups

In this article, we study symmetric $(v,k,\lambda )$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is ${\mathrm{PSU}}_4\left(q\right)$. We prove that there exist eight non-isomorphic such designs for which $\lambda \in \{3,6,18\}$ and $G$ is either ${\mathrm{PSU}}_4\left(2\right)$, or ${\mathrm{PSU}}_4\left(2\right):2$.     

Ree群~2G_2(q)与2-(v,k,l)区组设计(Ⅱ)

本文证明了自同构群的基柱为Ree群~2G_2(q)的区-本原2-(v,k,1)设计必为Ree unital,即2-(q~3+1,q+1,1)设计,从而部分地回答了Praeger问题.     

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.