On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds: (1) Every maximal subgroup of P is S-embedded in G. (2) Every cyclic subgroup H of P with prime order or order 4 (if P is a non-abelian 2-group and H Z∞(G)) is S-embedded in G.

ON F-z-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS

A subgroup H of a group G is called F-z-supplemented in G if there exists a subgroup K of G, such that G = HK and H∩K≤ Z∞F（G）, where Z∞F（G） is the F-hypercenter of G. We obtain some results about the F-z-supplemented subgroups and use them to determine the structure of some groups.

The Influence of Primitive Subgroups on the Structure of Finite Groups

A subgroup H of a group G is said to be primitive if it is a proper subgroup of the intersection of all subgroups of G containing H as its proper subgroup. The purpose of this note is to go further into the influence of primitive subgroups on the structure of finite groups. Some new results are obtained.

On Finite Groups that are the Product of Two Subnormal Supersoluble Subgroups

Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by B_(1)and B_(2),respectively.In this paper,a characterisation of groups in B_(1)or in B_(2)is given.By applying this new characterisation,some new properties of B_(1)(B_(2))and new proofs of many known results about B_(1)or B_(2)are obtained.Further,closure properties of B_(1)and B_(2)are discussed.

Large pore volume mesoporous aluminum oxide synthesized via nano-assembly

A new nano-assembly approach has been proposed for the preparation of macropore volume mesoporous aluminum oxide supports. Secondary nano-assembly and a frame structure mechanism for large pore volume mesoporous supports have been proposed. In a primary nano-assembly supersoluble micelle,aluminum hydroxide nanoparticles were precipitated in situ in surfactants with a volume balance (VB) less than 1,followed by secondary nano-assembly in linear and cylindrical shapes. The secondary nano-assembly of cylindrical aluminum hydroxides was calcined to form nano cylindrical aluminum oxides. For the formation of macropore volume mesoporous supports,we utilized a frame structure mechanism of mesoporous support,in which the exterior surface of the carrier may not be continuous. This macropore volume support has been used for the hydrotreatment of a residual oil catalyst,which possesses the following physical characteristics:pore volume 1.8―2.7 mL·g-1,specific surface area 180―429 m2·g-1,average pore diameter 17―57 nm,average pore diameter more than 10 nm (81%―94%),porosity 87%―93%,and crush strength 7.7―25 N·mm-1.

New characterizations of finite supersoluble groups

Let A be a subgroup of a group G and X a nonempty subset of G.A is called an X- semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T1 of T there exists an clement x∈X such that ATix=TixA.On the basis of this concept we obtain some new characterizations of finite supersoluble groups.

The influence of s-conditionally permutable subgroups on finite groups

A subgroup H of a group G is called s-conditionally permutable in G if for every Sylow subgroup T of G there exists an element x ∈ G such that HTx = TxH. Using the concept of s-conditionally permutable subgroups, some new characterizations of finite groups are obtained and several interesting results are generalized.

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.

S-semiembedded subgroups of finite groups

A subgroup H of a finite group G is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. We say that a subgroup H of a finite group G is S-semiembedded in G if there exists an s-permutable subgroup T of G such that TH is s-permutable in G and T ∩ H ≤ H-sG, where HsG is an s-semipermutable subgroup of G contained in H. In this paper, we investigate the influence of S-semiembedded subgroups on the structure of finite groups.

How does Diagonal Subgroup Embedding Determine the Structure of a Group?

Let G be a finite group.Let D={(g,g)|g∈C},the main diagonalsubgroup of G x G.In this paper,we consider the suitable generalized normalities orindex of D in G x G,some interesting results are obtained.

Groups with Cyclic Outer Automizers

We consider groups G such that NG（H）/HCG（H） is cyclic for all H ≤ G. More specifically, we characterise locally nilpotent and supersoluble groups with this property.

On Nearly σ-Embedded Subgroups of Finite Groups

Let G be a finite group,σ={σi|i∈ I}be some partition of the set of all primes and σ(G)={(σi|σi∩ π(G)≠0}.We say that a subgroup H of G is nearly cr-cmbcdded in G if there exists a cr-permutable subgroup T of G such that H∩T is a<r-permutable subgroup of G and H C\T<HaeG^where Haec is the subgroup of H generated by all those subgroups of H which are cr-permutably embedded in G.In this paper,we study the structure of G under the condition that some given subgroups of G are nearly cr-embedded in G.Some known results are generalized.

Groups with Supersoluble Non-normal Subgroups

The structure of groups in which many subgroups have a certain property X has been investigated for several choices of the property χ.In particular, groups whose non-normal subgroups are supersoluble are studied ia this paper. Moreover, groups with only finitely many normalizers of non-supersoluble groups are considered.

The Influence of s-Conditional Permutability of Subgroups on the Structure of Finite Groups

Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.

On X-Semipermutability of Some Subgroups of Finite Groups

Let A be a subgroup of a group G and X a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we try to use the X-semipermutability of some subgroups to characterize the structure of finite groups.