The Influence Of Max Subgroups On The Structure Of Finite Groups

A proper subgroup M of a finite group G is called a maximal subgroup if it is not contained in any proper subgroup of G.Maximal subgroups are a special type of subgroup in finite groups,which are important research objects in the field of finite groups and have a close relationship with the structure of finite groups,especially soluble groups,nilpotent groups,and supersolvable groups.With the continuous development of group theory,using the properties of maximal subgroups to study the structure of large groups has become a hot topic for many group theorists,and the number and generalized normalities of maximal subgroups can better describe the structure of large groups.This thesis will further explore the structure of finite groups based on the number and generalized normalities of maximal subgroups.Based on the normal and quantitative properties of maximal subgroups,this paper studies the influence of maximal subgroups on the structure of finite groups.The main research contents are summarized in the following two aspects:(1)Based on the normality and S-quasi-normality of maximal subgroups,the influence of maximal subgroups on the structure of finite groups is studied.First,by using the minimal counterexample method,the normality of maximal subgroups is analyzed and described,and the relevant conclusions of large groups are obtained when maximal subgroups are normal subgroups.In the same way,by analyzing the properties of s-quasi-normal subgroups of maximal subgroups,the solvability relationship between s-quasi-normal subgroups of maximal subgroups and finite groups is revealed.(2)Based on the number of maximal subgroups,the influence of maximal subgroups on the structure of finite groups is studied.By analyzing the number of maximal subgroups of nilpotent groups,the influence of the number of maximal subgroups on the structure of finite groups is discussed,and the specific group structure of finite nilpotent groups is shown.In the end,the conclusion and prospect are summarized,as well as the unsolved problems in the results of this paper,and new ideas are put forward for its improvement and expansion.