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COMMUTING ELEMENTS WITH RESPECT TO THE OPERATOR Λ IN INFINITE GROUPS
Rezaei, Rashid,Russo, Francesco G. Korean Mathematical Society 2016 대한수학회보 Vol.53 No.5
Using the notion of complete nonabelian exterior square $G\hat{\wedge}G$ of a pro-p-group G (p prime), we develop the theory of the exterior degree $\hat{d}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for $\hat{d}(G)$ with respect to d(G). Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the p-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on G and $\hat{d}(G)$.
Commuting elements with respect to the operator $\wedge $ in infinite groups
Rashid Rezaei,Francesco G. Russo 대한수학회 2016 대한수학회보 Vol.53 No.5
Using the notion of complete nonabelian exterior square $G \widehat{\wedge} G$ of a pro-$p$-group $G$ ($p$ prime), we develop the theory of the exterior degree $\widehat{\mathrm{d}}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs $\mathrm{d}(G)$. Among the main results of this paper, we describe upper and lower bounds for $\widehat{\mathrm{d}}(G)$ with respect to $\mathrm{d}(G)$. Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the $p$-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on $G$ and $\widehat{\mathrm{d}}(G)$.
SOME CONSEQUENCES OF THE EQUATION [x<sup>n</sup>, y] = 1 ON THE STRUCTURE OF A COMPACT GROUP
Erfanian, Ahmad,Rezaei, Rashid,Tolue, Behnaz Korean Mathematical Society 2013 대한수학회지 Vol.50 No.1
Given an integer $n{\geq}1$ and a compact group G, we find some restrictions for the probability that two randomly picked elements $x^n$ and $y$ of G commute. In the case $n=1$ this notion was investigated by W. H. Gustafson in 1973 and its influence on the structure of the group has been studied in the researches of several authors in last years.
COMMUTING POWERS AND EXTERIOR DEGREE OF FINITE GROUPS
Niroomand, Peyman,Rezaei, Rashid,Russo, Francesco G. Korean Mathematical Society 2012 대한수학회지 Vol.49 No.4
Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.
Commuting powers and exterior degree of finite groups
Peyman Niroomand,Rashid Rezaei,Francesco G. Russo 대한수학회 2012 대한수학회지 Vol.49 No.4
Recently, we have introduced a group invariant, which is re-lated to the number of elements x and y of a nite group G such that x ^ y = 1G^G in the exterior square G ^ G of G. This number gives re-strictions on the Schur multiplier of G and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form hm ^ k of H ^ K such that hm ^ k = 1H^K, where m 1 and H and K are arbitrary subgroups of G.
Some consequences of the equation [xn, y]=1 on the structure of a compact group
Ahmad Erfanian,Rashid Rezaei,Behnaz Tolue 대한수학회 2013 대한수학회지 Vol.50 No.1
Given an integer n≥1 and a compact group G, we find some restrictions for the probability that two randomly picked elements xn and y of G commute. In the case n=1 this notion was investigated by W.H.Gustafson in 1973 and its influence on the structure of the group has been studied in the researches of several authors in last years.
MACWILLIAMS IDENTITY FOR LINEAR CODES OVER FINITE CHAIN RINGS WITH RESPECT TO HOMOGENEOUS WEIGHT
Moeini, Mina,Rezaei, Rashid,Samei, Karim Korean Mathematical Society 2021 대한수학회보 Vol.58 No.5
In this paper, we obtain the MacWilliams identity for linear codes over finite chain rings with respect to homogeneous weight, and the product of chain rings.
Cyclic codes of length $p^s$ over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^e \rangle}$
Roghayeh Mohammadi Hesari,Masoumeh Mohebbei,Rashid Rezaei,Karim Samei 대한수학회 2024 대한수학회논문집 Vol.39 No.1
Let $R_e=\frac{\mathbb{F}_{p^m}[u]}{\langle u^e \rangle}$, where $p$ is a prime number, $ e $ is a positive integer and $u^e=0$. In this paper, we first characterize the structure of cyclic codes of length $p^s$ over $R_e$. These codes will be classified into $2^e $ distinct types. Among other results, in the case that $e=4$, the torsion codes of cyclic codes of length $ p^s $ over $ R_4$ are obtained. Also, we present some examples of cyclic codes of length $p^s $ over $R_e$.