本原置换群作用下区组设计的分类

This project is devote to the research of the block design theory and finite primitive permutation group theory. The aim of this study is try to classify some block designs with transitive properties by using the knowledge of permutation group theory. We will study the following two areas: (1). The classifaction of the flag-transitive 2-(v,k,λ) designs with help of primitve permutation group of almost simple type; (2). The classifaction of the symmetric partially balanced incomplete block designs (i.e. SPBIB designs) having some primitive automorphism group. We will use the classification theorem of finite simple groups to completely classify flag-transitive 2-(v,k,λ) designs which include symmetric designs and non-symmetric designs admitting an automorphism group G of almost simple type with alternating groups, classical groups, exceptional groups of Lie type and sprodic groups as their socle. Not only do the reseach of this project construct and classify some block designs using primitive group theory, but also it can understand some structures and properties of groups with the structures of block designs.

本项目是区组设计理论与有限本原置换群理论的交叉研究，旨在利用有限群理论对某些具有传递性质的区组设计进行分类。项目主要分以下两大块内容展开研究：(1).几乎单型本原置换群作用下的旗传递 2-(v,k,λ) 设计的完全分类；(2).某些本原自同构群作用下的传递的部分平衡不完全区组设计(也即SPBIB设计)的分类。其中，第一块内容要借助于有限单群分类定理，就自同构群的基柱是交错群、典型群、例外Lie型单群和散在单群等各种情形分别对对称设计和非对称设计两种类型的设计进行独立研究。本项目的研究既可以利用本原群的理论来构造和分类某些区组设计，又可以利用区组设计的结构来理解群的结构和性质。

完全分类了以散在单群M_{11}、M_{12}、J_1，交错群A_5为本原置换群基柱的旗传递点本原2-(v,k,λ)设计；研究了以PΩ^+_8(3)为基柱的旗传递点本原对称(v,k,λ)设计的分类问题。基本解决了以散在单群M_{23}、M_{24}、J_2，交错群A_6为本原置换群基柱的旗传递点本原2-(v,k,λ)设计的分类问题和某些单群作用下的SPBIB设计的分类问题。本项目的研究为2-设计的分类工作做出了一定的贡献，提供了一些实用的研究方法。