The Influence Of Some Subgroups On The Structure Of Finite Groups

In this paper,we have studied the influence of the properties of some subgroups on the structure of a finite group.It consists of four chapters.As an introduction of this paper,in the Chapter 1,we give a brief account of our main results.We also set out some basic concepts and theorems which are closely related to our results.Let G be a finite group.A number of authors have examined the structure of a finite group G,under the assumption that all minimal subgroups of G are well-situated in the group.N.It(?) showed that a group G of odd order is nilpotent provided that all minimal subgroups of G lie in the center of the group.A sharpened form of It(?)'s result is that G is p-nilpotent if every subgroup of order p is contained in the center Z(G) of G as p is odd and G is 2-nilpotent if every subgroup of order 2 or 4 is contained in the center Z(G) of G as p = 2.J.Buckley showed that if all minimal subgroups of an odd order group are normal,then the group is supersolvable.A.Yokoyama showed that:If(?) is a subgroup closed saturated formation,a finite solvable group G with quaternion-free Sylow 2-subgroups belongs to(?) provided that every minimal subgroup of G is contained in(?)-hypercenter of G.And so on.However ,all results being mentioned above are the sufficient conditions of some properties of finite groups.In order to explore the necessary and sufficient condition of p-nilpotent,p-supersolvable as well as(?)_p-group,we have investigated that the influence of minimal subgroups on the structure of a finite group under some conditions in Chapter 2.We consequently obtained some helpful results(see Proposition 2.2.1,Theorem 2.2.2,Corollary 2.2.3,Theorem 2.2.12,2.2.13,2.2.17,2.2.18,2.2.21,Corollary 2.2.22,Theorem 2.2.27, 2.2.30,2.2.33,2.2.35,Corollary 2.2.37),and improved and extended the works of other authors in this field.Let G be a finite group,its subgroups H and K are called permutable if = HK = KH.A subgroup H of the group G is called quasinormal (or permutable) in G if H permutes with all subgroups of G.Letπbe the set of prime divisors of the order of G,a subgroup H of G is calledπ-quasinormal (or S-quasinormal) in G if it permutes with every Sylow p-subgroup of G for each p inπ.Quasinormal andπ-quasinormal subgroups have many interesting properties.In Chapter 3 of this paper,we have discussed that the influence ofπ-quasinormality of 2-maximal subgroups of Sylow subgroups on the structure of a finite group,and some sufficient conditions are obtained for a finite group being p-nilpotent or to have an ordered Sylow tower property or supersolvable(see Theorem 3.2.2,3.2.5,3.2.8,3.2.10 ).Let G be a group,the norm of G,denoted by N(G),consists of all those elements of G which normalize every subgroup of the group.Clearly,N(G) is a characteristic subgroup of G and G is a Dedekind group if and only if G = N(G);and it is well-known that the center,denoted by Z(G),of G is contained in the norm N(G).This idea was introduced at first in 1935 by R.Baer,who delineated the basic properties of the norm.Up to now, many authors have investigated intermittently the property of N(G) and the relation between N(G) and the structure of G.In Chapter 4 of this paper, we have determined the presentation of finite groups with uncomplicated norm quotient.In Theorem 4.2.1 and Corollary 4.2.2,we have determined that the presentation of finite group with cyclic norm quotient;In Theorem 4.2.3 and Theorem 4.2.4,we have determined that the presentation of finite group with abelian norm quotient of cube-free order;In Theorem 4.2.5 and Corollary 4.2.8,we have determined that the presentation of finite group whose norm quotient are finite groups with cyclic Sylow subgroups and the finite groups with square-free order,respectively.In§4.3,we have defined the norm-center,norm-centralized embedded subgroup,Baer subgroup of a finite group G.Then,we have argued that the simple properties of these subgroups.For example,we have proved that the Fitting subgroup F(G) and the hyperquasicenter Q_∞(G) are contained in the Baer subgroup B(G) for every finite group G.