利用子群的嵌入性质、乘积和融合系的有限群的结构研究

给出具有特定性质子群的有限群的结构、子群积因子是如何影响大群的结构的等是有限群研究中的基本课题, 融合系是近年来从有限群及其表示发展起来的新理论. 本课题组将在从群系的观点出发，借助GAP等数学软件的帮助，研究一些具有特殊嵌入性质的子群(广义正规子群)的有限群的结构、给出具有特殊嵌入性质的子群(广义正规子群)本身的特征、讨论子群置换积等的内部结构、研究具有特殊子群链的群的结构, 进而用融合系这一范畴语言刻画有限群的结构. 课题组将强调刻画广义正规子群与正规子群、可补子群的距离的度量以及具有一定嵌入性质的子群的群的分类工作;将深入研究我们处于国内外领先的课题如置换积的因子对群结构的影响、具有特殊子群列的群结构方面；另外, 紧跟国际群论研究的潮流, 用有限群论局部分析的方法来研究融合系, 尝试用融合系这一范畴语言来刻画有限群的特征.

he following are basic subjects in the study of finite groups: giving thecstructure of finite groups with some subgroup having special properties, the influences of product factor of subgroups on the structure of groups. The fusion system is a new theory developed from finite groups and their representations in recent years. From the point of view of formation and with the help of mathematical software such as GAP, our team will study the structure of finite groups with some subgroups with special embedding properties (generalized normal subgroups), giving the characteristics of subgroups with special embedding properties (generalized normal subgroups), studying the internal structures of permutably products of subgroups and the structures of groups with special subgroup chains. Furthermore, the structure of finite groups is characterized by the category language of fusion system. We will emphasize the description of the metric of distance between generalized normal subgroups and normal subgroups, complemented subgroups and obtain the classification of subgroups with certain embedding properties. We will deeply study the influence of the factors of permutable product on the group structure, and the structure of groups with special subgroup sequences, which are our leading subjects at home and abroad. In addition, following the trend of international research on group theory, we use the technique of finite group theory local analysis to study the fusion system, and try to describe the characteristics of finite groups by using the category language of fusion system.