The Influences Of The Hall Embedded Properties Of Subgroups And The P-group Residual,Norm On The Structure Of Finite Groups

The content of this paper includes three aspects.The first,it is not only the most,principal way to study the structure of finite groups by its subgroups,but also an everlasting topic in the research of the finite groups.We call it the embedded properties of subgroups that the situation or the relation or the condition that the subgroups have in the whole group in brief.We call it the Hall embedded properties of subgroups if the subgroups can be the Hall subgroups of some subgroups.We introduce the concepts of Hall s-semiembedded subgroup and Hall conjugately embedded subgroup surrounded the Hall embedded properties of subgroups.We give some results about the structure of finite groups using these two concepts;The second,we discuss the influence of the norm on the structure of finite groups;The third,we consider the g-group residual Op(G)of a group G.We give the structure of finite groups using the p-subgroups with the same order.This paper is organized as the following five chapters.In Chapter 1,we introduce some notation,basic concepts and some results that we usually use in the paper.In Chapter 2.we discuss the influence of the Hall s-semiembedded subgroups on the structure of finite groups.Let G be a finite group,H a subgroup of G.If H is always a Hall subgroup of(H,P)for any prime p??(G)with(p,|H|)= 1,then H is called a Hall s-semicmbedded subgroup of G,where P? Sylp(G).We obtain some new characterizations for a finite group to be p-nilpotent and p-supersolvable or belong to an saturated formation with the assumption that some p-subgroups of the same order are Hall s-semiembedded subgroups,.In Chapter 3,we discuss the influence of the Hall conjugately embedded subgroups on the structure of finite groups.Let G be a finite group,H a subgroup of G.If H is always a Hall subgroup of<H,H9>for any g ? G,then H is called a Hall conjugately embedded subgroup of G.On one hand,We give some characterizations of finite groups with the assumption that some special subgroups are Hall conjugately embedded subgroups and give a new characterization of a solvable T-group.On the other hand,we also investigate the influence of Hall conjugately embedded subgroups on the structure of finite groups with the assumption that some subgroups of the Sylow p-subgroups with the same order are Hall conjugately embedded subgroups.Some characterizations for finite groups to be nilpotent,p-nilpotent and belong to an saturated formation are obtained.In Chapter 4,we discuss the influence of the Norm on the structure of finite groups.We obtain some results about the finite group G with the assumption that Norm(G)is maximal subgroup and 2-maximal subgroup of G respectively.Meanwhile,we also give some characterizations of a p-group G with the assumption that Norm(G)is 2-maximal subgroup of G.In Chapter 5,we discuss the influence of p-group residual OP(G)on the structure of group G.We mainly prove the following result by combining the p-subgroups with OP(G).Theorem.Let G be a finite group,p be a prime dividing the order of G,P ? Sylp(G),d be a power of p such that 1 ? d<|P|.Assume that H?OP(G)(?)Op(G)for all subgroups H(?)P with |H| = d,then either G is p-supersolvable,or else|P?Op(G)|>d.