有限p群的非正规子群的共轭类数与群结构的研究

令v(G)为有限p群G的非正规子群的共轭类数,其中p为素数。以往的研究只确定了闭区间[0,p+1]内v(G)的取值为0、1、p或p+1，并分别分类了v(G)取这些值的有限p群。我们实质性地推广了这些结果：确定了闭区间[p+2,2p]内v(G)的所有取值，并分类v(G)取定这些值的有限p群。以此为基础，本项目首先确定闭区间[2p+1,4p]内v(G)的所有取值，并分类v(G)取定这些值的有限p群。其次，找到有限3群的v(G)的取值规律。最后，对于固定的素数p，从各个角度探索有限p群及集合{v(G)|G为有限p群}的一般性质。

Let v(G) be the number of conjugacy classes of nonnormal subgroups of a finite p-group G，where p is a prime. Studier in the past have identified the value of v(G) is 0、1、p or p+1 and respectively given the complete classification of finite p-groups when v(G) belongs to the closed interval [0,p+1]. We essentially generalized these results: the value of v(G) is identified and all finite p-groups are completely classified when v(G) belongs to the closed interval [p+2,2p]. On the basis of these results, we will firstly identify the value of v(G) and give the complete classification of all finite p-groups when v(G) belongs to the closed interval [2p+1,4p]. Secondly, we will find out the rule of v(G) for finite 3-groups. Finally, for a fixed prime p, we will explore general characteristics of all finite p-groups and the set {v(G)|G is a finite p-group} from all perspectives.