The Influence Of The Number Of Orders Of Non-normal Subgroups On Finite Groups

In finite groups,using the properties of subgroups to study the structure and properties of groups is an important method.In this dissertation,we study the structure and properties of finite groups by using the number of orders of non-normal subgroups,which generalizes and enriches the study of non-normal subgroups.Let G be a finite group,J(G)denotes the number of orders of non-normal subgroups.This dissertation is divided into four chapters.In Chapter 1,we present the background of this dissertation and the main results contained in later chapters.In Chapter 2,we give some basic concepts and common conclusions involved in this dissertation.In Chapter 3,we mainly study finite groups with J(G)=2,prove that such groups are solvable,and give the group structure of such groups under non-nilpotent conditions.In Chapter 4,we mainly study solvable non-nilpotent groups with J(G)=t and prove that the derived length of G satisfies dl(G)≤[(2t+2)/3]+1.