The Influence Of Special Subgroups On The Structure Of Finite Groups

Maximal subgroup, minimal subgroup and abelian subgroup are three classes of very important subgroups, which played an important part in the study of the structure of finite groups. In the thesis, we study the structure of finite groups through considering the quantitative properties and normal properties of maximal subgroups, minimal subgroups and abelian subgroups, some new results are obtained. It consists of four chapters.In Chapter 1, we introduce some symbols and basic concepts which are used in the thesis.In Chapter 2, we mainly use the type of conjugacy classes of maximal subgroups to characterize the ordinary alternating groups, symmetric groups, solvable groups, non-solvable groups and finite groups which have a non-abelian simple subgroup, et al, some new results are obtained.In Chapter 3, we firstly give a class of non-solvable groups, which show that the hypothesis for Theorem 2 in [3] is very essential. Moreover, we give some examples to show that some hypotheses for a finite group to be p-nilpotent, solvable and supersolvable in some papers can not be removed, respectively. Secondly, some new generalizations for two important theorems in [3] are obtained. Lastly, a new result for solvable groups is given by considering the indices of maximal subgroups.In Chapter 4, we study finite groups whose the number of abelian subgroups are given. Then, we obtain a complete classification for finite groups which have exactly 5 or 6 non-trivial abelian subgroups, and we prove that finite groups which have less than 12 non-trivial abelian subgroups are solvable.