Lagrange定理、Sylow定理、可解群的Hall定理等结果给出了子群的存在性。子群的性质反过来决定了整个群的结构。利用子群的性质研究有限群的结构是有限群研究领域的一个热点课题。本项目研究σ-群理论与子群的置换性、嵌入性、可补性以及QCLT-群相关问题。具体研究问题是：1）将Sylow-子群的置换性、嵌入性推广和发展到某些(σ_i)-子群的置换性，由此来描述有限群的σ-可解性和σ-幂零性；2）将子群的可补性推广和发展到σ-群中，刻画有限群的分解结构；3）刻画不可分解的QCLT{2,3}-群；4）研究具有下列性质的有限群：任意素数p整除群G的阶|G|，对于G的满足$|B|_p=p$的子群B，G有|B|阶Hall-次正规嵌入子群；5）研究具有下列性质的有限群：对于G的任意子群B，G有|B|阶σ-置换嵌入子群。本项目的研究工作对揭示有限群的构造会产生积极的作用。

The fundamental theorems such as Lagrange Theorem, Sylow Theorem and Hall Theorem in soluble universe ensure the existence of some subgroups. Conversely, the properties of subgroups determine the global property of a finite group. It is an active research field to investigate the structure of groups by using the properties of subgroups. This project is to study the theory of σ-groups and the permutability, embedding property, complementation of subgroups, and some problems related to QCLT-groups. This project includes the following concrete research problems: 1) Extend the permutability and embedding of Sylow-subgroups to the permutability of some (σ_i)-subgroups, which will be used to describe σ-solubility and σ-nilpotent of finite groups; 2) Develop and extend the supplementation (complementation) of subgroups to the theory of σ-groups, and present the structural results of finite groups; 3) Characterize indecomposable QCLT {2, 3}-groups; 4) Study the finite groups G satisfying the following properties: for any prime p dividing |G|, and for any subgroup B which has Sylow p-subgroups of order p, G has a Hall subnormally embedded subgroup of order |B|; 5) Study the finite groups G satisfying the following properties: for any subgroup B, G has aσ-permutably embedded subgroup of order |B|. Our research work in this project will have positive impacts on understanding the structure of finite groups.