设G是一个有限群，G的norm(用N(G)表示）是G的每一个子群的正规化子的交。作为norm的推广，我们将研究有限群G的所有子群的中心的正规化子的交(用V(G)表示之)的性质对群结构的影响，以及一个有限群G的所有子群的超可解剩余或西洛塔剩余的正规化子的交(分别用WS(G)、WT(G)表示之)的性质对群结构的影响，并确定某些满足G＝V(G)，或G＝WS(G)，或G＝WT(G)的有限群G的结构。对于有限群G，集合Le(G)＝{x∈G|xe＝1}的大小与群G的结构有一定关系，其中e是G的阶｜G｜的任意一个正因子。对于正整数n，令b(n)为{ x∈G|xn＝1}中元素的个数，设c是大于1 的常数，函数f(n)＝cn2，我们将研究满足b(n)≤f(n)的群的结构性质与分类问题。

Let G be a finite group, the norm of G, denoted by N(G), is the intersection of the normalizers of all subgroups of G, we will consider some generalizations of norm. For example, we will study the influence of the properties of the subgroup V(G) which is the intersection of the normalizers of centers of all subgroups of G, and that of the intersection of the normalizers of the supersolvable residuals or the Sylow tower residuals of all subgroups of G (denoted by WS(G), WT(G), respectively). If we have G＝V(G)，or G＝WS(G)，or G＝WT(G), we will investigate the structures of G. Let G be a finite group and e be a positive integer dividing the order ｜G｜ of G, and write Le(G)＝{x∈G|xe＝1}, then there are some connections between the size of Le(G) and the structure of G. Given a positive integer n, let b(n) be the number of elements in the set { x∈G|xn＝1}, we will study the structural properties and the classifications of G such that b(n)≤f(n) for all divisors of the exponent of G, where f(n)＝cn2 and c is a constant such that c>1.