非正规子群链与有限p群结构的研究

In the study of finite p-groups, it is a basic method to characterize the structure of p-groups by the properties of all or some subgroups. People have achieved a lot of important research results by this simple and natural method. This method still plays a great role in the study of group theory. The propose of this project is to study the structure of finite p-groups and related problems by limiting nonnormal subgroups. The influence of the chain of nonnormal subgroups on the structure of finite p-groups and related problems is the main content of this project. We plan to study four problems: (1) the structure of finite p-groups whose length of chain of nonnormal subgroups is at most 3; (2) the structure of finite p-groups whose nonnormal subgroups are metacyclic (the rank of nonnormal subgroups at most 2); (3) the structure of finite p-groups whose length of chain of nonnormal subgroups is maximal; (4) determine whether it is the best to estimate the order of derived group, nilpotent class and the minimum number of generators by using the length of chain of nonnormal subgroup given by Passman. In addition, combining nonnormal subgroups with isolated subgroups to study the structure of p-groups is another content of this project, which will lay a certain foundation for the subsequent studies on isolated subgroups of finite p-groups.

在有限p群的研究中，通过全部或部分子群的性质来刻画p群自身的结构是一种基本方法。人们利用这个简单自然的方法取得了大量重要的研究成果。此方法至今仍在群论研究中发挥着巨大的作用。本项目旨在对非正规子群作一定限制来研究有限p群的结构以及相关的p群问题。特别地，非正规子群链对p群结构的影响及相关问题的研究是本项目的主要研究内容。我们拟研究四个问题：(1)非正规子群链长不超过3的有限p群的结构；(2)非正规子群均亚循环(秩不超过2)的有限p群的结构；(3)非正规子群链长最大的有限p群的结构；(4)判别Passman所给出的利用非正规子群链长对导群的阶、幂零类，最小生成元个数的估值是否最佳。另外，将非正规子群与孤立子群结合起来研究p群的结构是本项目的又一项研究内容，它将为后续关于有限p群的孤立子群的研究奠定一定的基础。

本项目主要研究非正规子群链对有限p群结构的影响及结合p群的孤立子群来研究p群的结构。首先，我们分类了chn(G)不超过2的有限p(p>2)群；分类了非正规子群的阶为p^{m-1}，p^m，p^{m+1}的有限p群；分类了非正规子群均具有循环极大子群的有限2群。其次，针对有限p群的孤立子群，研究了非正规子群均孤立的有限p群，给出了所有内交换子群均孤立的有限p群，具有亚循环孤立子群的有限p群以及具有交换孤立子群的有限p群的结构。 最后，还对亚交换的A_4群进行了刻画；给出了亚循环p群的p次中心扩张等。.本项目共发表学术论文4篇，其中2篇被SCI收录， 出版p群专著一部。另外还有一篇已被接受发表。总之，项目的预期目标圆满完成。