有限群的共轭类长及其相关问题研究

项目名称： 有限群的共轭类长及其相关问题研究

项目编号： No.11301378

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 孔庆军

作者单位： 天津工业大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 共轭类长；嵌入；自同构；导长；群作用

英文摘要： 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.

英文关键词： conjugacy class sizes；embedding；automorphism；derived length；action on groups