The Number Of The Orders Of Non-normal Subgroups And The Structure Of Finite Groups

Let G be a finite group. In this paper, we define J(G) to the number of the orders of non-normal subgroups, as and we study the relation between J(G) as the structure of the finite groups. We get some main results as follows:1. Let G be non-nilpotent. Then J(G)=1, if and only if G=[N]P is a spilt extension of a normal subgroup N of prime order q by a cyclic p-group P. Moreover, we have [N,Φ(P)]=1, and p＜q.2. If G is a nilpotent group and J(G)=1, then G a p-group.3. Let G be a metacyclic p-group. If J(G)=1, then one of the following holds:(ⅰ) G(?)M(p~n), where p is a prime, and n≥3if p is odd, and n≥4, if p=2;(ⅱ) G have a homomorph D(8);(ⅲ) G(?)Q(16).