arXiv subjects
Publications and source records attributed to Azam Hokmabadi.
In this paper, we determine the structure of the nilpotent multipliers of all pairs $(G,N)$ of finitely generated abelian groups where $N$ admits a complement in $G$. Moreover, some inequalities for the nilpotent multipliers of pairs of finite groups and their factor groups are given.
In this paper, we introduce the tensor analogues of commuting automorphisms and central automorphisms. Then we give several properties of such automorphisms and apply these new concepts to give some interesting results for $2_\otimes$-Engel elements.
Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$, with $|N|=p^n$ and $|G/N|=p^m$. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair $(G,N)$ of finite $p$-groups is bounded by $ p^{\frac{1}{2}n(2m+n-1)}$ and hence it is equal to $ p^{\frac{1}{2}n(2m+n-1)-t}$, for some non-negative integer $t$. Recently the authors characterized the structure of $(G,N)$ when $N$ has a complement in $G$ and $t\leq 3$. This paper is devoted to classify the structure of $(G,N)$ when $N$ has a normal complement in $G$ and $t=4,5$.
In this paper, we find an upper bound for the exponent of the Schur multiplier of a pair $(G,N)$ of finite $p$-groups, when $N$ admits a complement in $G$. As a consequence, we show that the exponent of the Schur multiplier of a pair $(G,N)$ divides $\exp(N)$ if $(G,N)$ is a pair of finite $p$-groups of class at most $p-1$. We also prove that if $N$ is powerfully embedded in $G$, then the exponent of the Schur multiplier of a pair $(G,N)$ divides $\exp(N)$.
In 1998, G. Ellis defined the Schur multiplier of a pair $(G,N)$ of groups and mentioned that this notion is a useful tool for studying pairs of groups. In this paper, we characterize the structure of a pair of finite $p$-groups $(G,N)$ in terms of the order of the Schur multiplier of $(G,N)$ under some conditions.
This article is devoted to present an explicit formula for the $c$th nilpotent multiplier of nilpotent products of some cyclic groups $G={\bf {Z}}\stackrel{n_1}{*}{\bf {Z}}\stackrel{n_2}{*}...\stackrel{n_{t-1}}{*}{\bf {Z}}\stackrel{n_{t}}{*} {\bf {Z}}_{m_{t+1}}\stackrel{n_{t+1}}{*}{\bf {Z}}_{m_{t+2}}\stackrel{n_{t+2}}{*}...\stackrel{n_{k}}{*}{\bf{Z}}_{m_{k+1}}$, where $m_{i+1} | m_i$ for all $t+1 \leq i \leq k$ and $c \geq n_1\geq n_2\geq ...\geq n_t\geq ...\geq n_{k}$ such that $ (p,m_{t+1})=1$ for all prime $p \leq n_1$. Moreover, we compute the polynilpotent multiplier of the group $G$ with respect to the polynilpotent variety ${\mathcal N}_{c_1,c_2,...,c_s}$, where $c_1 \geq n_1.$
In this article, we present an explicit formula for the $c$th nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c\geq 1$) of the $n$th nilpotent product of some cyclic groups $G={\mathbb {Z}}\stackrel{n}{*} ... \stackrel{n}{*}{\mathbb {Z}}\stackrel{n}{*} {\mathbb {Z}}_{r_1}\stackrel{n}{*} ... \stackrel{n}{*}{\mathbb{Z}}_{r_t}$, (m-copies of $\mathbb {Z}$), where $r_{i+1} | r_i$ for $1 \leq i \leq t-1$ and $c \geq n$ such that $ (p,r_1)=1$ for all primes $p$ less than or equal to $n$. Also, we compute the polynilpotent multiplier of the group $G$ with respect to the polynilpotent variety ${\mathcal N}_{c_1,c_2,...,c_t}$, where $c_1 \geq n.$
Let $G$ be a finite $p$-group of order $p^{n}$ with $|M(G)|=p^{\frac{n(n-1)}{2}-t},$ where $M(G)$ is the Schur multiplier of $G$. Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of $G$ when $t=0,1,2,3$. In this paper, we are going to find some structures for an abelian $p$-group $G$ with conditions on the exponents of $G, M(G),$ and $S_2M(G)$, where $S_2M(G)$ is the metabelian multiplier of $G$.
In this paper we present some inequalities for the order, the exponent and the number of generators of the $c$-nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c \geq 1$) of a finite group and specially of a finite $p$-group. Our results generalize some previous related results of M.R. Jones and M.R.R. Moghaddam. Also, we show that our results improve some of the previous inequalities.
The paper is devoted to finding a homomorphic image for the $c$-nilpotent multiplier of the verbal product of a family of groups with respect to a variety ${\mathcal V}$ when ${\mathcal V} \subseteq {\mathcal N}_{c}$ or ${\mathcal N}_{c}\subseteq {\mathcal V}$. Also a structure of the $c$-nilpotent multiplier of a special case of the verbal product, the nilpotent product, of cyclic groups is given. In fact, we present an explicit formula for the $c$-nilpotent multiplier of the $n$th nilpotent product of the group $G= {\bf {Z}}\stackrel{n}{*}...\stackrel{n}{*}{\bf {Z}}\stackrel{n}{*} {\bf {Z}}_{r_1}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{r_t}$, where $r_{i+1}$ divides $r_i$ for all $i$, $1 \leq i \leq t-1$, and $(p,r_1)=1$ for any prime $p$ less than or equal to $n+c$, for all positive integers $n$, $c$.
Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $\exp(G)$ and let $m=\lfloor\log_pk\rfloor$. We show that $\exp(M^{(c)}(G))$ divides $\exp(G)p^{m(k-1)}$, for all $c\geq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $\exp(M(G))$ divides $\exp(G)$ for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement of some previous bounds for the exponent of $M^{(c)}(G)$ given by M. R. Jones, G. Ellis and P. Moravec in some cases.