线性群轨道结构与有限群的算术性质

It is a main stream of the current research in group theory that using group representations, group actions and the corresponding orbit structures, and the arithmetic conditions of groups to study the group structures. The goal of this project is to make further progress along these lines and dig deeper, while at the same time to broaden the scope of problems tackled and methods used. The project plans to tackle several important open problems in finite group theory, including a conjecture of Morteo about the p-parts of character degrees, a conjecture of Gluck about the largest character degree and a conjecture of Thompson about recognizing simple groups using conjugacy class sizes. The PI has studied the existence of regular orbits of solvable primitive linear groups and obtained many novel results. The PI also has settled a conjecture of Espuelas-Carlips, and obtained the best known result with respect to the conjecture of Gluck about the largest character degree. The current project plans to use the orbit structure of linear group actions to tackle different problems, which is innovative...The main objectives of this project are as follows: .(1) Complete the classifications of solvable primitive linear groups which do not have regular orbits up to group and representation isomorphisms..(2) Study the relations of subgroups, quotient groups and orbit structures of linear groups..(3) Using group representations and the orbit structures to study the arithmetic conditions of conjugacy class sizes and character degrees..(4) Study the influence of the arithmetic conditions of orbit lengths, element orders, conjugacy class sizes and character degrees on the group structures.

有限群的表示、群作用的轨道结构以及群的算术性质研究是当前国际群论界的主流研究课题。本项目深化、扩展和完善已有的工作及方法，继续研讨群的轨道结构及群的算术性质。拟研究解决群论中的几个著名问题，包括Moreto有关特征标p部分的猜想，Gluck有关极大特征标的猜想以及Thompson有关共轭类刻画单群的猜想。申请人曾给出本原可解线性群轨道结构的深入刻画，解决了Espuelas-Carlips猜想并得到有关Gluck极大特征标猜想的已知的最好结果。本项目的创新点在于通过对线性群轨道结构的深入研讨来解决上述公开问题。主要研究内容为: （1）可解线性群正则轨道存在性的完全分类定理。（2）子群，商群及群轨道结构之间的联系。（3）用线性群轨道结构这一方法处理特征标和共轭类、元阶集等群的共轭不变量相关的算术性质的问题。（4）群作用轨道长度、元素的阶、特征标次数、共轭类的长度等对有限群结构的影响。

有限群的表示、群作用的轨道结构以及群的算术性质研究是当前国际群论界的主流研究课题。本项目深化、扩展和完善已有的工作及方法，继续研讨群的轨道结构及群的算术性质。对于群论中的几个著名问题进行了深入的研究，包括Moreto有关特征标p部分的猜想，Gluck有关极大特征标的猜想以及Thompson有关共轭类刻画单群的猜想。本项目在以下几个大方向进行了深入的研究，主要研究内容为: （1）可解线性群正则轨道存在性的完全分类定理。（2）子群，商群及群轨道结构之间的联系。（3）用线性群轨道结构这一方法处理特征标和共轭类、元阶集等群的共轭不变量相关的算术性质的问题。（4）群作用轨道长度、元素的阶、特征标次数、共轭类的长度等对有限群结构的影响。（5）用有限群的不变量刻画有限单群。整个项目组在本项目的支持下共完成论文41篇，其中发表在国际刊物上的有39篇，发表在SCI（SCIE）刊物上的有39篇，EI上1篇。