جهاد جمعة سليمان الجرادين

Throughout this paper, all groups are finite. Recall that a formation is a homomorphic F of groups such that each group G has a smallest normal subgroup (denoted by G F ) whose quotient is still in F. A formation F is said to be p-local if it contains each group G with G/(Φ(G) ∩ Op(G)) ∈ F. Our main goal here is to prove the following theorem

Let A,B be subgroups of a group G and ∅≠X⊆G. Then A is said to be X-permutable with B if there exists an element x∈X such that AB x =B x A. In this paper we study finite groups in which maximal subgroups of Sylow subgroups are X-permutable either with maximal subgroups or with Sylow subgroups.

The main target of this paper is to discuss whether the fuzzy folding of a fuzzy group is a fuzzy group or not. In addition, the relation between the fuzzy folding and the fuzzy retractions are obtained.AMS Mathematics Subject Classiftcation

The main aim of this article is to study some special elements (zero divisors and units) in ternary rings. Then, the main properties and concepts of noetherian and artinian ternary rings have been studied. In addition, new results on noetherian and artinian ternary rings have been investigated.

Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G.

Let G be a finite group. We say that a subgroup H of G is -normal in G if G has a subnormal subgroup T such that TH = G and (H ∩ T)HG/HG is contained in the -hypercenter of G/HG, where is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are -normal in G.

We investigate integral Galois stable representations of finite groups over local and global fields and their integers under the ground field extensions related to permutation modules. We consider normal extensions E/F and subgroups G ⊂ GLn(E) stable under the natural operation of the Galois group of E/F with some extra integrality conditions. The possible realization fields of G with these integrality conditions are of special interest, and we consider certain criteria of integrality for representations in GLn(E). We also study a series of related arithmetic problems and examples.

A subgroup H of a group G is called S-quasinormal in G if it permutes with every Sylow subgroup of G.The structure of the group G has been studied earlier by many authors under the assumption that the maximal or the minimal subgroups of the Sylow subgroups are well situated in G. In the present paper we are cincerned with the study of the structure of a finite group under the assumption that some subgroups of G are S-quasinormal in G, and we discuss some methods and applications.

A subgroup H of a group G is called S− quasinormal in G if it permutes with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of F∗(G) are S− quasinormal in G.

It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

Our aim in this paper is to prove some properties of semi-derivations centralizing on Jordan ideals of rings with involution. Some well known results characterizing commutativity of prime rings by derivations have been generalized by using semi-derivations. Moreover, we provide examples to show that our hypotheses are not superfluous.

In this article, we present some commutativity theorems for a ring R equipped with a generalized derivation satisfying certain differential identities on Jordan ideals of R. Some related results for prime rings are also discussed. Finally; we provide examples to show that the assumed restrictions are not superfluous.

A subgroup H of a group G is c-subnormal in G if G has a subnormal subgroup T such that HT=G and T 3 H ⊆ HG. [1] Using this concept, in Jaraden obtain [1] some new conditions for solubility of a finite group are given. Here we obtain local versions of these results.

Throughout this paper, all groups are finite. Recall that a formation is a homomorphic F of groups such that each group G has a smallest normal subgroup (denoted by G F ) whose quotient is still in F. A formation F is said to be p-local if it contains each group G with G/(Φ(G) ∩ Op(G)) ∈ F. Our main goal here is to prove the following theorem

Let A,B be subgroups of a group G and ∅≠X⊆G. Then A is said to be X-permutable with B if there exists an element x∈X such that AB x =B x A. In this paper we study finite groups in which maximal subgroups of Sylow subgroups are X-permutable either with maximal subgroups or with Sylow subgroups.

The main target of this paper is to discuss whether the fuzzy folding of a fuzzy group is a fuzzy group or not. In addition, the relation between the fuzzy folding and the fuzzy retractions are obtained.AMS Mathematics Subject Classiftcation

The main aim of this article is to study some special elements (zero divisors and units) in ternary rings. Then, the main properties and concepts of noetherian and artinian ternary rings have been studied. In addition, new results on noetherian and artinian ternary rings have been investigated.

Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G.

Let G be a finite group. We say that a subgroup H of G is -normal in G if G has a subnormal subgroup T such that TH = G and (H ∩ T)HG/HG is contained in the -hypercenter of G/HG, where is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are -normal in G.

We investigate integral Galois stable representations of finite groups over local and global fields and their integers under the ground field extensions related to permutation modules. We consider normal extensions E/F and subgroups G ⊂ GLn(E) stable under the natural operation of the Galois group of E/F with some extra integrality conditions. The possible realization fields of G with these integrality conditions are of special interest, and we consider certain criteria of integrality for representations in GLn(E). We also study a series of related arithmetic problems and examples.

A subgroup H of a group G is called S-quasinormal in G if it permutes with every Sylow subgroup of G.The structure of the group G has been studied earlier by many authors under the assumption that the maximal or the minimal subgroups of the Sylow subgroups are well situated in G. In the present paper we are cincerned with the study of the structure of a finite group under the assumption that some subgroups of G are S-quasinormal in G, and we discuss some methods and applications.

A subgroup H of a group G is called S− quasinormal in G if it permutes with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of F∗(G) are S− quasinormal in G.

It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

Our aim in this paper is to prove some properties of semi-derivations centralizing on Jordan ideals of rings with involution. Some well known results characterizing commutativity of prime rings by derivations have been generalized by using semi-derivations. Moreover, we provide examples to show that our hypotheses are not superfluous.

In this article, we present some commutativity theorems for a ring R equipped with a generalized derivation satisfying certain differential identities on Jordan ideals of R. Some related results for prime rings are also discussed. Finally; we provide examples to show that the assumed restrictions are not superfluous.

A subgroup H of a group G is c-subnormal in G if G has a subnormal subgroup T such that HT=G and T 3 H ⊆ HG. [1] Using this concept, in Jaraden obtain [1] some new conditions for solubility of a finite group are given. Here we obtain local versions of these results.