Researches And Applications On Some Generalized P-solvable Groups

Group is one of the basic algebraic systems in mathematics and finite group is the core of group theory.Exploring the structures and properties of groups is an important task in the study of finite groups.Any finite groups can be regarded as successive extensions of simple groups and simple groups.As the basic elements of group structure,simple groups play an important role in the study of finite groups.In this dissertation,we define a new class of generalized p-solvable groups,denoted by Sp*,based on the observation of some simple groups,the structural characteristics and quantitative information of Sylow subgroups,and inspired by the chief factor construction of p-solvable groups.For the class Sp*,we not only extend the cover-avoidance properties of subgroups to the generalized p-solvable coveravoidance properties of subgroups,but also extend the notion of F-subnormality to the notion of Sp*-subnormality.In this paper,we deeply study the construction of generalized p-solvable groups by giving different subgroup properties to maximal subgroups and second maximal subgroups,and also get a series of related applications.The main content of the paper is divided into the following four aspects.Firstly,the structure of the group class Sp*s characterized by the generalized psolvable cover-avoidance properties given to the corresponding maximal subgroups by limiting the number of maximal subgroups and combining with the idea of localization.In addition,general generalized p-solvable groups are characterized by giving generalized p-solvable cover-avoidance properties to Sylow subgroups and related subgroups.Secondly,consider the set of some maximal subgroups Maxp（G）,whose order is divisible by p.By classifying the corresponding second maximal subgroups and giving them the cover-avoidance properties,generalized p-solvable cover-avoidance properties,Sp*-subnormality properties to study the construction of generalized psolvable groups.Thirdly,consider the set of some maximal subgroups Maxη（G）,whose normal index is not a power of p.The structure of general generalized p-solvable groups is studied by classifying the corresponding second maximal subgroups and giving the generalized p-solvable cover-avoidance properties of subgroups.Fourthly,as applications,the class of p-solvable groups is characterized by the p-cover-avoidance properties of partial maximal subgroups and second maximal subgroups.And we describe the class of generalized p-solvable groups by using the core relation between the second maximal subgroups and the maximal subgroups containing them.