生成子群的性质与有限群结构的研究

To study the structure of finite groups is always one of the important subjects by the properties of subgroups in the finite group theory. The project under research mainly focus on the influence of the properties of the join of some subgroups on the structure of a finite group. In particular, we study the relationship between the join of a pair of conjugate subgroups and the structure of finite groups. Using the local analysis method, we investigate the solvable groups、p-supersolvable groups、p-nilpotent groups, etc. under the assumption that the properties and the sizes of the subgroups which are generated by two conjugate elements are well-situated. Next, we discuss the structure of the classes of groups which are more extensive than the LM-groups (the subgroup lattice is lower semi-modular). The research of this subject not only has significance for the solvable group theory, but also shows the useful applications of the Classification of Finite Simple Groups in the theory of solvable groups.

通过子群的性质来研究有限群的结构一直是有限群论中的重要课题之一。本项目主要研究生成子群的性质对有限群结构的影响，重点考查二共轭子群生成的子群与有限群结构的关系。本项目结合局部分析方法，从二共轭元生成的子群的固有性质和大小方面来讨论有限群的可解性、p-超可解性、p-幂零性等；讨论比LM-群(子群为下半模格的群)更加广泛的群类的性质及结构。这不仅对可解群论有重要的意义，而且体现了单群分类定理在可解群理论中的应用。

本项目结合局部分析方法，从一些临界群的精细结构入手，讨论了所有二共轭元生成子群的性质（p-超可解）及部分二共轭元生成子群的性质对群结构的影响；也考察了PLM-群的结构问题以及子群在其共轭子群中的广义正规性与群结构的关系。此外，还研究了子群的性质对群结构的影响。.本项目所研究的问题是可解群理论中的热点和难点课题，所获得的研究成果丰富了可解群理论。