有限p群的几个未解决问题的研究

Finite p-groups play an basic and important role in the theory of finite groups, the structrues and properties of finite groups are impacted by the structrues and properties of finite p-groups significatly. We are interested in the normal structrues of finite p-groups. In this program, we will classify the finite p-groups with some normal properties, which answer some problems introduced in “Groups of Prime Power Order”by Y.Berkovich. Furthermore, we rise and solve some new problems ，which generalize some problems of Y.Berkovich. The problems in this program are closely related to the normal structrues of finite p-groups, the normal structrues is very important in finite groups, especially in finite solvable groups.

有限p群是抽象有限群最基本和最重要的分支之一，有限p群的结构和性质对抽象有限群结构和性质的有重要影响。本项目主要从有限p群的正规结构入手，给出满足一定正规条件的有限p群的分类， 从而解决Y.Berkovich的p群专著《Groups of Prime Power Order》中提出的几个未解决问题，进一步提出并解决若干广义问题。其意义在于：该问题与p群的正规结构密切相关，而正规结构对于有限群，尤其是对有限可解群的性质和结构有重要影响。

本项目给出了非正规子群在其正规化子下的商群循环的有限 p 群的完全分类, 作为推论，也分类了非正规子群都是单列嵌入子群的有限p群, 这完全解决了Y. Berkovich 在其p群专著 《Groups of Prime Power Order Vol.1》中提出的Problem 134 和 Problem 156. 除此以外，我们还分类了以下几类p群：(1) 非正规子群的正规闭包较大的有限p群; (2) 非正规子群的正规核同阶的有限p群; (3) 交换非正规子群都循环的有限 p 群.