Researches On Some Classes Of Generalized Local Groups And Their Applications

Group theory is an important branch of algebra,while finite group theory is the core of the whole group theory.Finite simple groups are similar to primes in number theory,which play an irreplaceable role in the study of finite group theory.The completion of the classification theorem of finite simple groups is one of the most important achievements in the field of mathematics in the 20th century.One of the central tasks of finite group theory is to study the properties and structures of groups.In this dissertation,some classes of local groups including the class Gp*of generalized p-solvable groups,the class up#of generalized p-supersolvable groups and almost simple groups,etc.are studied and related applications are obtained.The dissertation is divided into five chapters.Chapter 1,The introduction.The research background and main results of the dissertation are introduced.Chapter 2,The basic definitions.Some basic definitions of group theory related to dissertation are given.Chapter 3,On the class Gp*of generalized p-solvable groups.The structure of the class Gp*of groups is characterized by the normal index of maximal subgroups,and then some applications in the class Gp*of groups are explored by means of G-boundary factors and G-traces of some specific subgroups.Chapter 4,On the class up#of generalized p-supersolvable groups.The struc-ture of the class up#of groups is characterized by the normal index of maximal sub-groups.In addition,by virtue of properties of the weakly M-supplemented primary subgroups,some applications in the class up#of groups are obtained,and then the internal relationship between the p-modular subgroups Op(G)and p-supersolva.ble groups is revealed.Chapter 5,The study of the weakly monomial subgroups of finite groups.By using the properties of the maximal subgroups of finite simple groups,the influ-ence of the weakly monomial subgroups on the structure of maximal subgroups is explored,and then some applications in almost simple groups are given.