有限单群和旗传递点本原对称设计

The classification of finite simple groups is one of the important results in group theory. Scholars are paying the growing attention to the research of the automorphism groups and classification of designs using the classification theory of finite simple groups, especially when the automorphism group of designs with certain good transitivity, e.g., point-transitive,block-transitive, flag-transitive, etc. . This project characterizes the internal connections between the theory of flag-transitive symmetric designs and the theory of finite groups. Under certain condition, the automorphism group of symmetric designs can be reduced to an affine or almost simple group by using of the O'Nan-Scott theorem and the classification theorem of finite simple groups. We focus on the almost simple case. First, we characterize the classification results of the symmetric designs when the socle of its automorphism group is the alternating group. Then, we study the classification results of the symmetric designs with classical socle.

有限单群的分类是群论中的重要结果之一。利用有限单群分类定理来研究设计的自同构群并将设计进行分类受到了越来越多学者的关注，尤其是当设计的自同构群具有某些良好的传递性（如：点传递、区传递、旗传递等）时。. 本项目研究的是在旗传递条件下的对称设计理论与有限群理论之间的内在联系。在某些特定的条件下，依据 O'Nan-Scott 定理和有限单群分类定理，对称设计的自同构群可归约到仿射型和几乎单型上来。我们关心的就是几乎单型的情况。首先，考虑在对称设计自同构群的基柱为交错群的情况下对设计的完全分类。然后，考虑其基柱为典型群时的对称设计的完全分类结果。

近年来，具有某些对称性质的组合设计理论和有限群理论的内在联系引起了广泛关注。研究设计的某些对称性，如：点传递、旗传递，可以有助于我们更好的理解一些群的结构。. 本项目研究的是 lambda=4 时的旗传递 (v, k, lambda) 对称设计。在某些特定的条件下，根据O'Nan-Scott 定理和有限单群分类定理，对称设计的自同构群可以归约到仿射型和几乎单型。我们关心的就是几乎单型的情况。本项目讨论了对称设计自同构群的基柱为典型群时的对称设计的分类。当基柱为 PSLn(q) 时，得到 2-(15,8,4) 对称设计且基柱为 PSL2(9)。