可解线性群的正则轨道及应用

线性群理论是沟通群论和表示论的桥梁。最近，研究有限线性群的轨道已成为了这两个领域中的重要课题之一。因为这是解决许多有重要意义的有限群论问题的关键工具，比如 Gluck-猜想，Karmonikov问题等。. 在这个项目中，我们将研究有限可解线性群的大(正则)轨道的存在性问题并利用这些轨道理论的方法和结果尝试去解决（或部分解决）一些重要的相关问题和猜想。我们将主要关注研究可解本原线性群和一些非本原的情形，如果可能的话，我们将尝试地去研究一般的情形。我们相信如果这个项目有重大的进展，它将给我们提供一个有效的途径去解决许多有趣的问题并更好的去理解有限可解群。

The theory of linear groups is a bridge between group theory and representation theory. Recently，it has become one of the most important topics in these two fields to investigate the orbits of finite linear groups. Because it is a key tool to deal with many important problems in finite group theory, such as Gluck’s conjecture, Karmonikov’s problems and so on.. In this project, we will study the existence of large(regular) orbits of finite solvable linear groups and try to solve some problems in an orbit-theory way. We will mainly focus on the solvable primitive linear groups and some imprimitive cases, and if possible, we will try to extend to the general case. We believe if this project has a significant development, it will offer an effective way for us to solve many interesting problems and to understand finite solvable groups better.