有限群的共轭类长及其相关问题研究

The applicant investigates the influence of conjugacy class sizes and the number of conjugacy class sizes of some special elements on the structure of finite groups by using the research method that is from the local to the whole, and combining with action on groups, embedding, automorphisms and so on, in order to reveal the interdependence between conjugacy class sizes, the number of conjugacy class sizes, automorphisms and action on groups, and offer a new approach for the research of this aspect. As the deepening of research, the applicant tries to establish relationship between conjugacy class sizes of some special elements of finite groups and Fitting length and derived length of finite groups, so as to promote the research of some well-known conjecture, especially Huppert's conjecture and Berkovich's conjecture. At last, as an application, the applicant studies conjugacy class graphs of finite groups. The results of this project not only have a very important theoretical significance for the development of finite group theory, but also play a positive role in promoting the development of other related disciplines.

申请人采用从部分到整体的研究方法，通过群作用、嵌入以及自同构等技巧研究有限群的某些特殊元素的共轭类长和共轭类长的个数对有限群结构的影响，以此来揭示有限群的共轭类长，共轭类长的个数，自同构以及群作用之间的相互依存关系，为这方面的研究提供一种新的研究途径。作为研究的深化，试图建立有限群某些特殊元素的共轭类长与有限群的Fitting长和导长之间的关系，从而推动某些相关著名猜想的研究，特别是Huppert猜想和Berkovich猜想。最后，作为应用，研究有限群的共轭类图。本项目的研究成果不仅对有限群论的发展有着十分重要的理论意义，而且还会对其他相关学科的发展起到积极的推动作用。

本项目按申请书计划进行，完成了项目预期目标，共发表论文16篇，其中SCI索引10篇， EI索引1篇。共轭类长是有限群的一个重要数量性质，在刻画有限群的结构方面有很多的应用。本项目主要是利用素数幂阶或双素数幂阶元素的共轭类长的数量性质来研究有限群的结构及其相关问题。取得的主要研究成果为：（1）对有限群中素数幂阶元素和双素数幂阶元素的共轭类长的个数为3时群的结构进行了研究，给出了这类群的详细结构；证明了当有限群中素数幂阶元素和双素数幂阶元素的共轭类长的个数为4时，群的可解性和幂零性；（2）利用素数幂阶元素，给出了有限群中一个元素的共轭类长为素数幂的几个充分条件；（3）构造了一个新的共轭类的子图，刻画了有限群的结构。