项目组成员施武杰教授于上世纪80年代开始首先研究群的数量刻画，对"群的阶"和"元的阶之集"以及单独对"元的阶之集合"刻画有限群做过长期研究，目前前一种刻画已经完成。陈贵云在上世纪90年代研究Thompson猜想，即共轭类长度集合对有限单群的刻画研究。B. Huppert提出关于群的特征标维数集合对群的刻画，并做了研究。.在本项目中第一方面研究减弱数量限制来刻画有限群，即：弱条件下的有限群的数量结构四个问题研究 问题一 有限群高阶元素的阶和群的阶对群的结构的影响；问题二 特殊共轭类长度与群结构；问题三 用群的阶及高维不可约特征标的维数刻画单群； 问题四 单群的素图顶点度数型对单群结构的影响。其次研究《Finite p-groups》的公开问题12.2.15和《Groups of Prime Power Order》(卷1)的公开问题148；研究几种Dedekind群的推广形式。

A member of this project Wujie Shi has been studying quantitative characterization of finite groups since early 1980's by using "order of a group" and "the set of element-orders" or only "the set of element-orders", at the present the previuos characterization of finite simple groups was finished. Guiyun Chen has been studying Thompson's conjecture since 1990, that is, characterizing finite simple groups by using the set of lengths of conjugacy classes. He proved the Conjecture holds for finie groups with non-connected prime graphes. B. Huppert conjectured that every finie simple group can uniquely determined by the set of dimensions of irreducible characters and proved the conjecture holds for some finite groups. .In this project, we first focus on weaking restrictions of quantities, which contains four questions of quantitative characterizations on weaked restrictions of quantities: 1.How do the maximal orders of elements and group order influence the stucture of a finite group; 2. How do the lengths of special conjuagcy classes influence the stucture of a finite group; 3. How do maximal dimensions and the group order influence the stucture of a finite group; 4. How do the degrees of vertex of prime graph of a finite group influence its structure.Secondly, we study open question 12.2.15 in 《Finite p-groups》 and open question 148 in 《Groups of Prime Power Order》(volume 1); several kinds of generalizations of Dedekind groups.