关于有限p-群子群正规化子和正规核的一些问题研究

有限p-群是有限群的一个重要群类.随着单群分类的最终完成,很多群论科学家开始转向有限p-群的研究.因为有限p-群的结构相当复杂,故其一般同构分类问题是十分困难的.于是利用子群的正规化子和正规核来研究有限p-群的结构成为重要的研究课题之一. 相关关注和研究很多,例如:Berkovich在其有限p-群专著中提出的116个问题是关于正规化子较小的研究问题; Cutolo,Smith,Wiegold等研究了正规核较大的n核p-群;徐明曜教授在其p-群专著中把p核p-群列为研究专题,等等.. . 本项目主要从子群的正规化子和正规核的角度出发,应用群论局部分析的思想以及p-群扩张理论研究有限p-群结构.首先研究正规化子较小的有限p-群的性质和结构.其次,研究并给出正规核较小的有限p-群的结构表达式.最后,在得到结果的基础上进一步探索研究正规化子、正规闭包以及正规核之间的本质联系.

Finite p-groups is an important part of finite groups. With the completion of the classification of the finite simple group, many scientists in group theory began to study finite p-groups. Finite p-groups is so complicated that it is quite difficult to give a complete classification of all non-isomorphic p-groups. So to study p-groups by using the normalizers and normal cores of the subgroups becomes one of topics in finite p-groups. There are many related concerns and researches. The Problem 116 proposed by Berkovich in his book of finite p-groups is about normalizers; Cutolo, Smith and Wiegold have studied n-core p-groups with larger normal cores; Professor Xu Mingyao has listed p-core p-groups as research topics in his monograph on p-groups; and so on.. In this project, we investigate the structure of finite p-groups from the point of view of normalizers and normal cores of subgroups by applying the idea of local analysis and the theory of extension of finite p-group. Firstly, we study the finite p-groups which have small normalizers. Secondly, we investigate the structure of finite p-groups when the normal cores of some subgroups are small. Finally, we will do further exploration and research to reveal the intrinsic link between normalizers, normal closures and normal cores on the basis of the results obtained.