(α,γ)-Anti-Multi-Fuzzy Subgroups and Some of Its Properties

Recently,fuzzy multi-sets have come to the forefront of scientists’interest and have been used in algebraic structures such asmulti-groups,multirings,anti-fuzzy multigroup and(α,γ)-anti-fuzzy subgroups.In this paper,we first summarize the knowledge about the algebraic structure of fuzzy multi-sets such as(α,γ)-anti-multi-fuzzy subgroups.In a way,the notion of anti-fuzzy multigroup is an application of anti-fuzzy multi sets to the theory of group.The concept of anti-fuzzy multigroup is a complement of an algebraic structure of a fuzzy multi set that generalizes both the theories of classical group and fuzzy group.The aim of this paper is to highlight the connection between fuzzy multi-sets and algebraic structures from an anti-fuzzification point of view.Therefore,in this paper,we define(α,γ)-antimulti-fuzzy subgroups,(α,γ)-anti-multi-fuzzy normal subgroups,(α,γ)-antimulti-fuzzy homomorphism on(α,γ)-anti-multi-fuzzy subgroups and these been explicated some algebraic structures.Then,we introduce the concept(α,γ)-anti-multi-fuzzy subgroups and(α,γ)-anti-multi-fuzzy normal subgroups and of their properties.This new concept of homomorphism as a bridge among set theory,fuzzy set theory,anti-fuzzy multi sets theory and group theory and also shows the effect of anti-fuzzy multi sets on a group structure.Certain results that discuss the(α,γ)cuts of anti-fuzzy multigroup are explored.

Research on θ -Intuitionistic Fuzzy Homomorphism Theory

Fuzzy homomorphism is an important research content of fuzzy group theory, different fuzzy mappings will produce different fuzzy homomorphisms. In this paper, the fuzzy homomorphism of groups is generalized. Firstly, the θ-intuitionistic fuzzy mapping is defined, and the θ-intuitionistic fuzzy homomorphism of groups is obtained. The properties of intuitionistic fuzzy subgroups and intuitionistic fuzzy normal subgroups are studied under the θ-intuitionistic fuzzy homomorphism of groups, and the fundamental theorem of θ-intuitionistic fuzzy homomorphism is proved.

Homomorphic Properties of Fuzzy Rough Groups

This paper is devoted to the discussion of homomorphic properties of fuzzy rough groups.The fuzzy approximation space was generated by fuzzy normal subgroups and the fuzzy rough approximation operators were discussed in the frame of fuzzy rough set model.The basic properties of fuzzy rough approximation operators were obtained.

FINITE p-GROUPS WHICH CONTAIN A SELF-CENTRALIZING CYCLIC NORMAL SUBGROUP

For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.

CHARACTERIZATION OF MODULAR FROBENIUS GROUPS OF SPECIAL TYPE

In this article, we first investigate the properties of modular Frobenius groups. Then, we consider the case that G＇ is a minimal normal subgroup of a modular Frobenius group G. We give the complete classification of G when G＇ as a modular Frobenius kernel has no more than four conjugacy classes in G.

Finite p-Groups with a Class of Complemented Normal Subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in de（G） have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified. Keywords Finite p-groups, normal subgroups, subgroup complement

On Weakly Normal Subgroups of a Finite Group

A subgroup H of a group G is said to be weakly normal in G if Hg ≤ NG（H） implies that g E NG（H）. In this paper, using the condition that the minimal subgroups or subgroups of prime square order of G are weakly normal in G, we get some results about formation.

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.

On Local Nilpotency of the Normal Subgroups of a Group

A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.

On Isomorphism Testing of Groups with Normal Hall Subgroups

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur- Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall （STACS 2009）, we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential （in the number of coordinates） time algorithm for code isomorphism problem developed recently by Babai et al. （SODA 2011）. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups： given two representations ρandτ- of a group H over Zp^d, a prime, determine if there exists an automorphism ： ФH→ H, such that the induced representation p Ф= ρ o Ф and τ are equivalent, in time poly（｜H｜, p^d）.

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.

Finite Groups Whose Nontrivial Normal Subgroups Have Order Two

In this paper, we investigate the structure of the groups whose nontrivial normal subgroups have order two. Some properties of this kind of groups are obtained.

Finite Groups Whose Nontrivial Normal Subgroups Have the Same Order

In this paper, we study finite groups all of whose nontrivial normal subgroups have the same order. In the solvable case, the groups are determined. In the insolvable case, some characterizations are given.

On Coprime G-conjugacy Class Sizes in a Normal Subgroup

Let N be a normal subgroup of a group G. Suppose that the positive integers m 〉 n are two longest non-central G-conjugacy class sizes of N with （m, n） = 1. The purpose of this paper is to determine the structure of N and give the N-conjugacy class sizes of the elements in N under that assumption that m is square free.

Finite Groups in Which Every Subgroup Is Abelian or Normal

The main object of this paper is to investigate the finite groups in which every subgroup is either abelian or normal. We obtain a characterization of the groups for the nonnilpotent case, and we also give some properties for the nilpotent case.

Thompson's conjecture for Lie type groups E_7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type ET（q）. More precisely, we show that every finite group G with the properties Z（G） = 1 and cs（G） = cs（ET（q）） is necessarily isomorphic to ET（q）, where cs（G） and Z（G） are the set of lengths of conjugacy classes of G and the center of G respectively.

On the Normal Subgroup with Exactly Two G-Conjugacy Class Sizes

Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.

Properties of A Class of Groups on Zm

In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_z^y） is solvable.We also prove that group GL（n,Z_p）is solvable if and only if GL（n,Z_p） is solvable,and list the generators of groups GL（n,Z_p） and SL（n,Z_p）.At last,we prove that PSL（2,Z_p）（ p〉3） and PSL（n,Z_p） （ n〉3） are simple.

On Conjugacy Class Graph of Normal Subgroup

Let G be a finite group and N a normal subgroup of G.Denote by Γ_(G)(N)the graph whose vertices are all distinct G-conjugacy class sizes of non-central elements in N,and two vertices of Γ_(G)(N)are adjacent if and only if they are not coprime numbers.We prove that if the center Z(N)=Z(G)∩N and Γ_(G)(N)is k-regular for k≥1,then either a section of Nis a quasi-Frobenius group or Γ_(G)(N)is a complete graph with k+1 vertices.