| The influence of the properties of subgroups on the structure of finite groups is one of the important topics in the study of finite group theory.This paper mainly study the structure of finite groups by the properties of minimal normal subgroups,and give some new results based on the known results.Firstly,we introduce the research background of this subject,and list some useful concepts and related theorems closely related to this paper.Then we study the structure of finite groups by using the properties of minimal normal subgroups,and obtain some results.We study the influence of the c-normal of minimal normal subgroups on the structure of finite groups.When the minimal normal subgroup of group G satisfies c-normal,a sufficient condition for the supersolvability of a group is given by using the relationship between nilpotent residue and quaternion group.At the same time,a sufficient condition for a group to be supersolvable when the minimal normal subgroup of the normal subgroup N of group G satisfies c-normal and the normal subgroup N is independent of the quaternion group are given.The results are extended from c-normal to s-normal,and a sufficient condition for the nilpotence of finite groups is obtained through the hypercenter Z∞(G)Secondly,using the concept of complementary group,we study the relationship between the minimal normal subgroup and the nilpotent hypercenter Z∞(G)of G when the four-order cyclic subgroup of finite group G or the four-order cyclic subgroup of normal subgroup is complementary.Some sufficient conditions for the nilpotence of the group when the minimal normal subgroup is included in the nilpotent hypercenter Z∞(G)are given and extended to p-nilpotent.Finally,we study the influence of the complete conditional permutation of minimal normal subgroups on the structure of finite groups.When the minimal normal subgroup of normal subgroup N of group G is completely conditionally permutation in G,some sufficient conditions for the supersolvability of finite groups are obtained by using Sylow 2-subgroups,which are extended to the hypercenter Z∞(G),and a sufficient condition for the nilpotence of finite groups is obtained.Finally,the work of this paper is summarized and the research results of this paper are prospected. |
