极小非Σ群的结构及其Cayley图的正规性

One of the hottest topics in finite group theory is to determinate the structure of groups which have a group theoretical property. Let Σ be a group theoretical property. If all proper subgroups of a group G have the property Σ but G does not have it, then G is called a minimal non-Σ-group. It often plays a key role in investigating the structure of finite groups. In this project, using the idea of local analysis, we will unify and generalize some properties of subgroups by some techniques, such as permutability, covering, embedding, etc. Then we will investigate the structure of finite groups by applying these new Σ properties. Furthermore, we will give classifications of some minimal non-Σ-groups. As applications of finite group theory on algebraic graph theory, first we will determine the full automorphism groups of Cayley graphs on minimal non-abelian groups combining the property of minimal non-abelian groups and the composite structure of graphs. Next we will study the normality of this kind of graphs and give its classification as complete as possible. At the end, we will consider the normality of Cayley graphs on other minimal non-Σ-groups. Our results will enrich finite group theory and algebraic graph theory.

确定具有某种群论属性的有限群的结构是有限群论热点课题之一。在解决这类问题中，极小非Σ群（若群G的每个真子群为Σ群，但G本身不是Σ群，其中Σ是一种群性质）的结构往往起到关键作用。本项目运用局部分析的方法，通过置换、覆盖以及嵌入等技巧统一并推广子群的Σ性质，再利用某些子群具有新的Σ性质来刻画有限群的结构，进而研究几类广泛的极小非Σ群的结构。作为有限群论在代数图论上的应用，我们结合极小非交换群的群论性质与图的组合结构，决定极小非交换群上的Cayley图的全自同构群，再研究这类图的正规性，并尝试给出其分类。在此基础上，进一步探索其它类别的极小非Σ群上的Cayley图的正规性。本项目的研究成果将丰富有限群论和代数图论的内涵。

确定具有某种群论属性的有限群的结构是有限群论热点课题之一。在解决这类问题中，极小非Σ群（若群G的每个真子群为Σ群但G本身不是Σ群，其中Σ是一种群性质）的结构往往起到关键作用。本项目运用局部分析的方法，通过置换、覆盖以及嵌入等技巧统一并推广了子群的Σ性质，利用某些子群具有新的Σ性质刻画了有限群的结构，进而确定出几类更广泛的极小非Σ群的结构。作为有限群论在代数图论上的应用，我们利用全自同构群的子群结构和正规商图理论，证明了不存在8p和8p^2阶三度半对称图。我们还研究了量子信息中量子混合态的局域等价分类问题；利用Heisenberg-Weyl观测基表达量子态，然后给出了一个两体量子态纠缠的判据，并把它推广到多体情形。本项目的研究成果丰富了有限群论和代数图论的内涵。