有限群数量性质是有限群一个重要的研究内容, 通过数量性质的研究可以更好地了解群的结构。同阶型是有限群的一个重要的数量。1987年J.G. Thompson提出运用同阶型来判断群的可解性的问题（简称Thompson问题），现已取得一定进展。本项目深化、扩展和完善已有的工作，进一步研讨Thompson问题及与其相关的问题。研究内容的主要创新点为：（1）Thompson问题在素图非连通时，已经被申请人等证明，在素图连通时怎么样处理？同阶型与其它数量有什么联系（比如Sylow 数、元素的阶、 同阶元个数、共轭类长度等数量)？(2) 有限群Sylow数研究, 侧重非交换单群的Sylow数及可解Sylow数的研究。（3）研究元素阶集合、同阶元个数集合刻画群，及同阶型与共轭类长度的联系。本项目试图用特征标理论及代数表示论的一些新思想来找到有限群数量结构研究的新方法，最终解决Thompson 问题。

The quantitative property is an important research area in the finite group theory, which is studied in order to know the structure of finite groups well. The same order type of a finite group is an important quantity. In 1987, J.G. Thompson posed a problem whether the solubility of finite groups is judged by the same order type (shorten by "Thompson Problem"). By far, it makes some progress. This project aims to extend the previous results. Also we continue to discuss Thompson Problem and related problems. Innovative points of this research mainly include: (1) how to deal with the case of groups with connected prime graph? But the applicant and professor Wujie Shi confirmed Thompson Problem in the case of groups with disconnected prime graph. Furthermore, we will study the relation between the same order type and other quantities such as Sylow numbers, orders of elements, the number of elements of the same order and the size of conjugacy class, etc. (2) we will research Sylow numbers of finite groups, and emphasis on the research of Sylow numbers of non-abelian simple groups and solvable Sylow numbers. Moreover, we will study the following problem. Let S be a non-abelian simple group and denote by SN(G) the set of Sylow number of G. If G has a trivial center and SN(G)=SN(S), whether or not G is isomorphic to S ? (3) we will study the characterization of finite groups by the set of orders of elements and of numbers of elements of the same order, and research the relation between the same order type and the size of conjugacy. This project tries to resolve Thompson Problem through some new methods that is originated from some new ideas of the character theory and the presentation theory of algebras.