子群正规性的对偶问题及其有限群的结构

有限群的结构始终是有限群论的核心研究课题之一，而正规子群在有限群结构研究中扮演着非常重要的角色。对偶地，非正规子群对有限群结构的影响同样起着关键作用。.本项目旨在借助部分非正规子群的性质，应用局部分析的思想，充分利用反证、归纳等方法和技巧刻画有限群的结构，揭示有限可解群和有限p群的本质，为这方面的研究提供一种新的研究途径。拟解决以下具体问题：确定有限p群G的非拟正规子群的共轭类个数u(G)在闭区间[1，tp]（其中t是较小整数，p是奇素数）内的所有取值，并分类u(G)取定这些值的有限p群；给出部分非正规子群具有特定群论性质时有限群的结构；利用部分非正规子群的个数及其共轭类个数分别刻画有限群的结构；确定以上结构中一些相应临界群的分类。.本项目所研究的内容处于该领域的前沿，我们的研究成果将丰富并推进有限群论的进展。

The structure of finite groups has always been one of the core research topics of finite group theory, and normal subgroups play a crucial role in the study of finite groups structure. Dually, the influence of non-normal subgroups on the structure of finite groups is also enormous. .The purpose of this project is to characterize the structure of finite groups and reveal the nature of finite solvable groups and finite p-groups under the assumption that partial non-normal subgroups satisfy some properties, by applying the idea of local analysis and making full use of methods and techniques such as proof by contradiction and induction. Consequently, this project will provide a new approach for research in this area. The project intends to solve the following specific issues: determine all values of the conjugate class number u(G) of non-quasinormal subgroups of finite p-groups G, and classify these groups for all values of u(G), where u(G)∈[1, tp], t is a smaller integer and p is an odd prime; present the structure of finite groups whose partial non-normal subgroups admit certain group-theoretic properties; characterize the structures of finite groups by applying the number of partial non-normal subgroups and the number of their conjugate classes respectively; and determine the classifications of corresponding critical groups in the above structures..The topics studied in this project are at the forefront of this field, and our achievements will enrich and advance the progress of finite group theory.