The Influence Of Some Special Second Maximal Subgroups On The Structure Of Finite Groups And Their Applications

The study of finite group theory can be roughly divided into solvable groups and non-solvable groups.With the completion of the classification theorem of finite simple groups,Wielandt once pointed out that the study of finite groups should give priority to the research methods and technical means of solvable groups to more general groups classes.In this dissertation,following this guiding ideology,we are committed to using the groups classes theory to infiltrate the research ideas of solvable groups into the study of non-solvable groups especially some simple groups.In particular,a series of new maximal subgroups are constructed from the perspective of duality properties of subgroups,and the corresponding second maximal subgroups are classified relying on the cores relations,subgroup properties,and the relations between them and some groups classes are revealed.On this basis,appropriate subgroup properties are given to investigate their impact on the structure of finite groups.The main content of this dissertation is divided into five chapters:In Chapter one,the research background and main results related to this dissertation are introduced.In Chapter two,some basic concepts and lemmas involved in this dissertation are given.In Chapter three,the solvable groups are described by classifying the second maximal subgroups with the aid of core relations.In addition,some new characteristics of solvable groups are given by using the cover and avoidance properties.As applications,the above classification localization is applied to explore the basic properties of generalized p-local groups classes.In Chapter four,second maximal subgroups are classified by means of index properties,and their cross classification with nilpotent and strong are considered.Firstly,the structure of correlation groups classes are revealed by their existence.Secondly,by means of core relations,boundary factor properties and cover and avoid properties,we promote the related research of generalized p-local groups classes.In Chapter five,subgroup properties are used to classify second maximal subgroups.Similarly,on the premise that the classifications are nonempty,appropriate subgroup properties are given to second maximal subgroups satisfying some properties to reveal the structure of the related generalized p-local groups classes.