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.

F-Decomposition of Minimax Module over Hyper Finite Groups

Let G be a hyper finite locally solvable group, A a minimax ZG-medule, a locally defined formation consisting of locally solvable groups, A has no nonzero infinite irreducible ZG-factors, and G ∈ . The following results are proved： if A has a maximal submodule B such that A/B is , central in G and B has no nonzero central ZG-factors, then A has an decomposition; ifA has an irreducible central submodule B such that all ZG-composition factors of A/B are o^eccentric, then A has an decomposition.

Weak Isomorphism of Finite Group

On the basis of the quasi-isomorphism of finite groups, a new mapping, weak isomorphism, from a finite group to another finite group is defined. Let G and H be two finite groups and G be weak-isomorphic to H. Then G≌H if G satisfies one of the following conditions. 1） G is a finite Abelian group. 2） The order of G is p^3. 3 ） The order of G is p^n＋1 and G has a cyclic normal subgroup N = 〈a〉 of order p^n. 4） G is a nilpotent group and if p^││G│, then for any P ∈ Sylp （G）, P has a cyclic maximal subgroup, where p is a prime; 5） G is a maximal class group of order p4（p〉3）.

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.

Finite Groups Isospectral to Simple Groups

The spectrum of a finite group is the set of element orders of this group.The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum,in particular,to list all finite simple groups for which the recognition problem is solved.

On Distance Laplacian Spectra of Certain Finite Groups

For a finite group G,the power graph P(G)is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if x^(i)=y or y^(i)=x,for 2≤i,j≤n.In this paper,we obtain the distance Laplacian spectrum of power graphs of finite groups such as cyclic groups,dihedral groups,dicyclic groups,abelian groups and elementary abelian p groups.Moreover,we find the largest and second smallest distance Laplacian eigenvalue of power graphs of such groups.

Character codegrees in finite groups

For an irreducible character x of a finite group G,we define its codegree as cod(x)=|G:ker x|/x(1) .In this paper,we introduce some known results x(1)and unsolved problems about character codegrees in finite groups.

Dynamical Properties of Additive Cellular Automata over Finite Abelian Groups

Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.

Local Embeddings of Some Families of Subgroups of Finite Groups

In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.

FINITE GROUPS WHOSE AUTOMORPHISM GROUP HAS ORDER CUBEFREE

LetGdenote a finite group.It is shown that if|Aut(G)|is cubefree thenGpossesses the Sylow tower property.

On Boundary Factors and Traces of Subgroups of Finite Groups

A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.For any G-boundary factor H/AG of A,we call the subgroup(A∩H)/AG of G/AG a G-trace of A.On the basis of these notions,we give some new characterizations of hypercyclically embedded subgroups.

On-Abnormal Maximal Subgroups of Finite Groups

For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.

Some Sufficient Conditions on the Number of Non-abelian Subgroups of a Finite Group to be Solvable

Thompson＇s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson＇s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.

On σ-semipermutable Subgroups of Finite Groups

Let a = {σi｜ i ∈ I} be some partition of the set of all primes P, G a finite group and σ（G） = {σi｜σi ∩ π （G） ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ（G）. A subgroup H of G is said to be： σ-semipermutablc in G with respect to H if HHi x = Hi x H for all x ∈ G and all x ∈ G and all Hi ∈H such that （｜H｜, ｜Hi｜） = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.

Finite Groups with H-Permutable Subgroups

Letσ={σi|i∈I}be some partition of the set P of all primes and G afinite group.A set H of subgroups of G is said to be a complete Hallσ-set of G ifevery member≠1 of H is a Hallσi-subgroup of G for some i c l and H containsexactly one Hallσi-subgroup of G for every i such thatσi∩π(G)≠Ø.A subgroupA of G is said to be H-permutable if A permutes with all members of the completeHallσ-set H of G.In this paper,we study the structure of G under the assuming thatsome subgroups of G areσ-permutable.

Finite Groups with Seminormal Sylow Subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.

A Note on TI-Subgroups of a Finite Group

Let G be a finite group and H a subgroup of G. Recall that H is said to be aTI-subgroup ofG ifHg∩H = 1 or H for each b∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G are TI-subgroups, then G is soluble, and all non-nilpotent subgroups of G are normal.

Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles

Motivated by Problem 164 proposed by Y. Berkovich and E. Zhmud＇ in their book ＂Characters of Finite Groups＂, we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character codegree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs （associated with degrees） have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.

A Note on Zeros of Characters of Finite Groups

The aim of this note is to classify the finite meta-abelian groups in which every irreducible character vanishes on at most three conjugacy classes.

Special Blocks of Finite Groups

We first determine in this paper the structure of the generalized Fitting subgroup F＊ （G） of the finite groups G all of whose defect groups （of blocks） are conjugate under the automorphism group Aut（G） to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/Op（L） has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O2（L） is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.