The Influence Of Some Special Subgroups On Generalized P-solvable Groups

Conducting a comprehensive and in-depth study on the structure of various groups is one of the most important part in group theory.The characteristics of subgroups,especially the supplemented properties,is an effective method to study the structure of groups.In this paper,our research is mainly divided into two parts.Firstly,we use the concept of the normal index of maximal subgroups to study the class of generalized p-solvable groups deeply and system-atically,combined with the properties of the maximal prime factor and the index of maximal subgroups in a finite group G.Secondly,we give two special supplemented properties of gen-eralized p-solvable groups.one is to impose a subnormal restriction on the supplemented subgroups,the other is to restrict the perspective of the hypercenter to subgroups.Using the properties of(?)p*-supplement and(?)p*-hypercentral supplement of Sylow subgroup’s maxi-mal subgroups to study the construction of the class of generalized p-solvable groups(?)p*.At the same time,on the basis of the class of generalized p-solvable groups(?)p*,a new group class(?)p2*is given too.The corresponding properties of supplemented subgroups are defined to study the structure of this group class.This thesis is divided into three parts:The first part is preliminary knowledge,which mainly gives the related concepts and lemmas involved in this article.The second part mainly studies the influence of the normal index of maximal subgroups on the class of generalized p-solvable groups(?)p*,Combined with the properties of the maximal prime factor and the index of the maximal subgroups in group G.The main results are as follows:(1)Let G be a finite group and p an odd prime dividing the order of G.Then G ∈(?)p*if and only if(η(G:M))p≤p for every maximal subgroup M∈Fp(G)={M | M<·G and |G:M|p=1}.(2)Let p be the largest prime dividing the order of G and P a Sylow p-subgroup of G.Then G ∈(?)p*if and only if(η(G:M))p≤p for every maximal subgroup M ∈ Fpc(G)={M<·G | M(?)NG(P)and |G:M| is composite}.The third part mainly uses the newly defined supplemented properties to study the struc-ture of related group class.Here we only list the results of the class of generalized p-solvable groups(?)p*.(1)Let G be a finite group and P a Sylow p-subgroup of G,where p is an odd prime dividing the order of G,then G ∈(?)p*if and only if every maximal subgroup of P has a(?)p*-supplement in G.(2)Let G be a finite group and P a Sylow p-subgroup of G,where p is an odd prime dividing the order of G,then G ∈(?)p*if and only if every maximal subgroup of P has a(?)p*-hypercentral supplement in G.