有限置换群与组合设计

In this project, we will study the action of finite permutation groups on combinatorial designs, and investigate the properties and classification of combinatorial designs by the group action. Concretely, we will study the following problems: 1. Research on flag-transitive 2-(v,k,λ) symmetric designs. The research can be divided into two parts, the first part is on the existence of point-primitive symmetric designs by using the O'Nan-Scott Theorem of the classification of finite primitive permutation groups;the second part is on point-quasiprimitive symmetric designs which relay on the classification of qusi-primitive permutation groups. The main aims are to reduce this type of designs to affine or almost simple type and then discuss their existences and classification. Furthmore, we also study the existence and classification of the symmetric designs with λ is small and the automorphism groups are point-imprimitive. 2. Research on block-transitive 2-(v,k,1) designs. We focus mainly on the Camina-Neumann-Praeger Project on the classification of block-transitive designs,especially on the affine type designs, the calssification of block designs with small parameters, and the Delandtsheer Conjecture on block-primitive designs. Our research is based on finite primitive and quasiprimitive permutaion groups, and the aim is the classification of such type of designs. The studies are related and influenced each other，and they are also challenging.

本项目主要探讨有限置换群与组合设计,通过群作用来研究组合设计的性质和分类。具体地,我们将研究以下几个方面的问题: 一) 研究旗传递的2-(v,k,λ)对称设计,其一是点本原的情形,涉及将设计的自同构群归约到仿射和几乎单群的一个猜想以及几乎单时的分类；其二是点拟本原的情形,利用拟本原置换群来分析此类设计的存在性和分类问题；其三研究非点本原λ较小时设计的存在性和结构。 二) 研究区传递的2-(v,k,1)设计,集中在区传递设计的Camian-Neumann-Praeger分类纲领上,特别是自同构群是仿射型时区传递设计的存在性和分类，具有较小参数或点数无平方因子的设计的分类,以及与自同构群区本原有关的Delandtsheer猜想。这些研究以本原和拟本原置换群的理论为基础,以对称设计和2-(v,k,1)设计的分类为目的,它们之间相互联系又互相影响。

项目研究设计的自同构群与分类。本项目首先研究了旗传递点本原的2-(v,k,λ)对称设计，得到了在某些条件下，比如(r, λ)=1, 或对一般的λ，基柱是交错群或某些例外李型单群设计的分类；$λ\geq(r, λ)^2$时，自同构群的归约问题、分类问题；若自同构群是旗传递点拟本原的，得到当λ=2,3时的归约定理。 得到了点数为两个素数之积的点本原2-(v,k,1)设计的分类，以及自同构群为极端本原群的2-(v,k,1)设计的部分结果。项目还得到了旗传递点本原非对称2-设计的有关结果，如基柱为交错群A_n且(r, λ)=1时只有3个设计，将λ=2,3时归约为仿射群或几乎单群的情形，并获得有关分类。此外，在区传递2-设计方面也得到部分结果。项目共发表论文36篇，其中SCI论文26篇，其它论文9篇。项目以本原和拟本原置换群的理论为基础,以对称（非对称）设计和2-(v,k,1)设计的分类为目的,它们之间相互联系又互相影响，这些成果极大地丰富了群与设计理论。

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