群的数量性质及其应用

Recently, significant progress has been achieved in the research of the quantitative property of finite simple groups. However, the generalized permutability of subgroups of simple groups has generally received little attention. In this project, we shall study the quantitative property of groups and its applications in the discussion of the generalized permutability of subgroups of the simple groups by applying the Classification of the Finite Simple Groups and the results in number theory and graph theory. We focus on the following topics:. 1. The influence of the special conjugacy class sizes on the structure of almost simple groups is studied and the alternating groups and their automorphism groups are characterized by their orders and special conjugacy class sizes. This research is related to a conjecture of Thompson about the conjugacy class sizes of simple groups.. 2. The quantitative property of simple groups is employed in the studying of the X-permutability of subgroups of the alternating groups. This study pertains to the open problem 17.37 in 《Unsolved Problems in Group Theory》 (18th edition) published by Siberian Division of Russian Academy of Sciences.. We shall study the simplicity of finite groups in view of their generalized permutable subgroups together with their quantitative properties, which is innovative and explorative. In addition, the project is expected to be innovative in theory, method and technology.

目前，有限单群的数量性质的研究在国内外都有了较大的进展。但是，关于单群中子群的广义置换性质方面的研究成果极少。本项目利用有限单群分类定理和数论、图论中的相关成果，研究群的数量性质及其在单群的广义置换子群研究中的应用。主要研究内容：1.特殊共轭类长对几乎单群结构的影响，用特殊共轭类长和群的阶刻画交错群及其自同构群，该研究与J.G.Thompson的一个关于共轭类长的猜想有关；2.利用单群的数量性质，研究交错群中子群的X-可置换性，该研究与俄罗斯科学院西伯利亚分院出版的《Unsolved Problems in Group Theory》（第18版）中的公开问题17.37有关。本项目利用子群的X-可置换性并结合群的数量性质来判断群的单性，在研究内容上是一个创新和探索。本课题预期在理论、方法和技术上有所创新。

有限群论是数学的一个重要分支，关于群的数量性质及其应用的研究是当前国际上较为热门的一个研究领域。本项目的研究属于这一领域。在本项目中，我们主要研究了有限单群的共轭类长刻画，以及单群中的条件置换子群。我们得到如下的研究成果。首先，利用群的阶和元素最高阶刻画了某些特殊的PGL(2, q)，这里q为一个素数或者为2^n且2^n+1或者2^n-1为素数。我们也给出例子说明，有些PGL(2, q)并不能有其阶和元素的最高阶来确定，例如PGL(2, 9)。其次，证明了交错群A_p，这里p为任一大于等于5的素数，可由其阶和共轭类长（p-1）！/2来刻画。这一结论的证明主要依赖有限单群分类定理，特别是素图不连通的单群的分类。第三，讨论了素数幂阶子群的X-s-半置换性与群结构之间的联系，对有限群的超可解性和p-幂零性给出了一些新的判别准则。本文的另一个主要结果为确定了所有子群都是G-s-半置换的有限群的结构，对这样的群给出了清晰的描述。最后，证明了群PGL(2, p)可由其阶和某个特殊的共轭类长来刻画，即：设G为一个群，那么G同构于PGL(2, p)当且仅当两者的阶相同且G有一个共轭类的长度为（p^2-1）。上述研究有助于人们理解和认识交错群的数量性质和子群性质。