低维有限典型群与线传递2-(v,k,1)设计

The research about 2-(v,k,1) designs and their automorphism groups is an important problem between groups and combinatorial designs. In this project, we will study the actions of finite permutation groups on 2-(v,k,1) designs, and then investigate the existence and classification of 2-(v,k,1) designs. Concretely, the following problems will be concerned..On the one hand, research the finite classical groups of dimension at most 24, and then classify the 2-(v,k,1) designs whose automorphism groups are n-dimensional finite simple classical groups and line-transitive, here n is no more than 24;.On the other hand, research the finite primitive permutation groups of square-free degree, and then classify the 2-(v,k,1) designs admitting.line-transitive point-imprimitive automorphism groups and with number of points being square-free..Our research is based on the theory of finite groups and the aim is to classify the 2-(v,k,1) designs.

对2-(v,k,1)设计及其自同构群的研究已经成为群与设计的一个重要课题。本项目通过研究有限置换群在2-(v,k,1)设计上的作用来讨论2-(v,k,1)设计的存在与分类问题。具体的，我们将研究以下两方面的内容：.（一）研究维数不超过24的有限典型群，分类自同构群是n-维有限典型单群（n≤24）的线传递2-(v,k,1)设计；.（二）研究次数无平方素因子的有限本原群，分类点数无平方素因子的线传递点非本原的2-(v,k,1)设计。.本项目是以有限群理论为基础，以分类2-(v,k,1)设计为目的的研究。

有限置换群的研究促进着2-(v,k,1)设计的研究进程。本项目致力于2-(v,k,1)设计及其自同构群的研究。在对维数不超过24的有限典型群和次数无平方素因子本原群的研究过程中，我们分类了点数为两个素数乘积的点本原2-(v,k,1)设计和点数为两个素数乘积的线传递点非本原的2-(v,k,1)设计，为后续研究奠定了坚实的基础。