Ruhollah Barzegar - Academia.edu (original) (raw)

Uploads

Papers by Ruhollah Barzegar

Research paper thumbnail of Probability Of Mutually Commuting Two Finite Subsets Of A Finite Group

Ars Comb., 2016

For a finite group GGG let P(m,n,G)P(m,n,G)P(m,n,G) denote the probability that two mmm-subset and nnn-subset o... more For a finite group GGG let P(m,n,G)P(m,n,G)P(m,n,G) denote the probability that two mmm-subset and nnn-subset of GGG commute elementwise and let P(n,G)=P(1,n,G)P(n,G)=P(1,n,G)P(n,G)=P(1,n,G) be the probability that an element commutes with an nnn-subset of GGG. Some lower and upper bounds are given for P(m,n,G)P(m,n,G)P(m,n,G) and it is shown that P(m,n,G)m,n\{P(m,n,G)\}_{m,n}P(m,n,G)m,n is decreasing with respect to mmm and nnn. Also P(m,n,G)P(m,n,G)P(m,n,G) is computed for some classes of finite groups, including groups with central factor of order p2p^2p2 and P(n,G)P(n,G)P(n,G) is computed for groups with central factor of order p3p^3p3 and wreath products of finite abelian groups.

Research paper thumbnail of On fundamental isomorphism theorems in soft subgroups

Soft Computing, 2000

Molodsov initiated a novel concept of soft set theory, which is a completely new approach for mod... more Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory.

Research paper thumbnail of On fundamental isomorphism theorems in soft subgroups

Molodsov initiated a novel concept of soft set theory, which is a completely new approach for mod... more Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory.

Research paper thumbnail of Nilpotency and solubility of groups relative to an automorphism

In this paper we introduce the concept of α-commutator which its definition is based on generaliz... more In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism α are defined. N (G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphisms of the group, respectively. If G is nilpotent or solvable with respect to the all its automorphisms, then is referred as it absolute nilpotent or solvable group. Subsequently, N (G) and S(G) are obtained for certain groups. This work is a study of the nilpotency and solvability of the group G from the point of view of the automorphism which the nilpotent and solvable groups have been divided to smaller classes of the nilpotency and the solvability with respect to its automorphisms.

Research paper thumbnail of Finite groups with three relative commutativity degrees

Bull. Iranian Math. Soc. 32(2) (2013), 271 - 280

For a finite group GGG and a subgroup HHH of GGG, the relative commutativity degree of HHH in GGG... more For a finite group GGG and a subgroup HHH of GGG, the relative commutativity degree of HHH in GGG, denoted by d(H,G)d(H,G)d(H,G), is the probability that an element of HHH commutes with an element of GGG. Let mathcalD(G)=d(H,G):HleqG\mathcal{D}(G)=\{d(H,G):H\leq G\}mathcalD(G)=d(H,G):HleqG be the set of all relative commutativity degrees of subgroups of GGG. It is shown that a finite group GGG admits three relative commutativity degrees if and only if G/Z(G)G/Z(G)G/Z(G) is a non-cyclic group of order pqpqpq, where ppp and qqq are primes. Moreover, we determine all the relative commutativity degrees of some known groups.

Research paper thumbnail of Probability Of Mutually Commuting Two Finite Subsets Of A Finite Group

Ars Comb., 2016

For a finite group GGG let P(m,n,G)P(m,n,G)P(m,n,G) denote the probability that two mmm-subset and nnn-subset o... more For a finite group GGG let P(m,n,G)P(m,n,G)P(m,n,G) denote the probability that two mmm-subset and nnn-subset of GGG commute elementwise and let P(n,G)=P(1,n,G)P(n,G)=P(1,n,G)P(n,G)=P(1,n,G) be the probability that an element commutes with an nnn-subset of GGG. Some lower and upper bounds are given for P(m,n,G)P(m,n,G)P(m,n,G) and it is shown that P(m,n,G)m,n\{P(m,n,G)\}_{m,n}P(m,n,G)m,n is decreasing with respect to mmm and nnn. Also P(m,n,G)P(m,n,G)P(m,n,G) is computed for some classes of finite groups, including groups with central factor of order p2p^2p2 and P(n,G)P(n,G)P(n,G) is computed for groups with central factor of order p3p^3p3 and wreath products of finite abelian groups.

Research paper thumbnail of On fundamental isomorphism theorems in soft subgroups

Soft Computing, 2000

Molodsov initiated a novel concept of soft set theory, which is a completely new approach for mod... more Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory.

Research paper thumbnail of On fundamental isomorphism theorems in soft subgroups

Molodsov initiated a novel concept of soft set theory, which is a completely new approach for mod... more Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory.

Research paper thumbnail of Nilpotency and solubility of groups relative to an automorphism

In this paper we introduce the concept of α-commutator which its definition is based on generaliz... more In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism α are defined. N (G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphisms of the group, respectively. If G is nilpotent or solvable with respect to the all its automorphisms, then is referred as it absolute nilpotent or solvable group. Subsequently, N (G) and S(G) are obtained for certain groups. This work is a study of the nilpotency and solvability of the group G from the point of view of the automorphism which the nilpotent and solvable groups have been divided to smaller classes of the nilpotency and the solvability with respect to its automorphisms.

Research paper thumbnail of Finite groups with three relative commutativity degrees

Bull. Iranian Math. Soc. 32(2) (2013), 271 - 280

For a finite group GGG and a subgroup HHH of GGG, the relative commutativity degree of HHH in GGG... more For a finite group GGG and a subgroup HHH of GGG, the relative commutativity degree of HHH in GGG, denoted by d(H,G)d(H,G)d(H,G), is the probability that an element of HHH commutes with an element of GGG. Let mathcalD(G)=d(H,G):HleqG\mathcal{D}(G)=\{d(H,G):H\leq G\}mathcalD(G)=d(H,G):HleqG be the set of all relative commutativity degrees of subgroups of GGG. It is shown that a finite group GGG admits three relative commutativity degrees if and only if G/Z(G)G/Z(G)G/Z(G) is a non-cyclic group of order pqpqpq, where ppp and qqq are primes. Moreover, we determine all the relative commutativity degrees of some known groups.