The Applications Of The Localization Of Properities Of Subgroups On The Related Group Classes

One of the key task of Group Theory research is to illustrate the properties and structure of various group classes.Finite Group Theory is the fundamental part of Group Theory,and based on the character of Finite Group Theory,researchers have proposed many kinds of definitions of group classes.Among practical research-es regarding group classes,group formation theory is widely accepted and applied by many researchers,which normally include saturated formation and solvable saturat-ed formation.In this paper,we will investigate the influence of several generalized M-supple-mented subgroups on the the structure of finite groups,which not only show that the structure of saturated formation such as solvable groups,p-nilpotent groups and p-supersolvable groups et al,and the structure of solvable saturated formation such as p-quasisupersolvable groups et al,but study the structure of the formation which is not saturated by the characterization of simple groups.We divide the dissertation into five chapters.Chapter 1,The introduction.We introduce the background.Chapter 2,The basic definition and lemma.We introduce some basic definitions and lemmas of The Finite Group Theory in this dissertation.Chapter 3,The influence of Mp-supplemented subgroups on the structure of group classes.First,we introduce a new group class T including all p-supersolvable groups and some non-solvable groups,and which is not saturated formation for subgroup closed.Second,we investigate the influence of the properties of Mp-supplemented subgroups on composition series and chief series and the structure of the group formation T.At the same time,we study the structure of finite groups by using p-modular subgroups.Last,we study the structure of solvable saturated formation such as p-quasisupersolvable groups by using Mp-supplemented subgroups.Chapter 4,The influence of some generalized M-supplemented subgroups on the structure of group classes.We study the structure of the group formation T and saturated formation and solvable saturated formation by considering some generalized M-supplemented primary subgroups.Chapter 5,The influence of localized generalized M-supplemented subgroups on the structure of finite groups.We study the structure of saturated formation un-der the assumption that prime order subgroups are generalized M-supplemented in normalizers of Sylow subgroups,and which focuses on a given prime order general-ized M-supplemented subgroup,and combine properies of some special subgroups.