对称设计与本原自同构群

The classification of flag-transitive design is a typical problem under the interaction between group and combination, and it has become one of the forefront subject of the finite groups theory and combinatorial designs theory. As a part of this project, the research on flag-transitive (v,k,λ)-symmetric designs is already in full swing. After the classification of flag-transitive (v,k,λ)-symmetric designs with λ small was completed, we have the ambition to completely conquer the question about the classification of flag-transitive (v,k,λ)-symmetric designs with λ general. Based on this aim, in this project we will use the classification theorem of primitive groups to prove that if D be a (v,k,λ)-symmetric design which admits a flag-transitive, point-primitive automorphism group G, then G must be of almost simple or affine type. And then, we will study the action of primitive group of almost simple type on symmetric designs, and use the classification theorem of finite simple groups to completely classify flag-transitive (v,k,λ)-symmetric designs admitting a point-primitive automorphism group G of almost simple type with alternating group or some of classical groups or exceptional groups of Lie type as its socle.

对称设计的分类问题是群与组合设计相互作用的一个典型问题，目前已经成为了有限群论和组合设计理论研究的一个前沿课题。作为其子工程, 对旗传递的(v,k,λ)-对称设计的研究工作正在紧张而有序地进行之中。当参数λ较小时的旗传递对称设计的分类完成之后，我们有信心攻克一般λ情况下的旗传递对称设计的分类问题。本项申请就是基于这个目标，拟利用本原群分类定理验证旗传递(v,k,λ)-对称设计的本原自同构群只能是仿射型或几乎单型的，并在此基础之上，研究几乎单型本原群在对称设计上的作用，拟利用有限单群分类定理，给出设计的自同构群的基柱是交错群及部分典型群、例外Lie型单群情形下的旗传递点本原(v,k,λ)-对称设计的完全分类。

有限群在某些组合结构，尤其是区组组合设计领域有着重要的应用价值。本项目致力于研究借助于本原群分类定理给出自同构基柱为部分几乎单型本原群的旗传递点本原(v,k,λ)-对称设计的完全分类及某些对称的部分平衡不完全区组设计的分类问题。业已完成了散在单群、低阶交错群An及某些典型群或例外李型群，如PSL(2,q）,PSL(12,2)作为本原自同构群基柱的设计的分类问题，以及以PGL(2,7)和M_{12}作为本原自同构群的SPBIB设计的分类问题，为后续研究奠定了坚实的基础。