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Behrooz Mashayekhy

Publications and source records attributed to Behrooz Mashayekhy.

At least 73 records · Page 4Linked to original sources
arXiv2011

Polynilpotent Multipliers of Some Nilpotent Products of Cyclic Groups

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 $c_1 \geq n.$

math.GR↗
arXiv2011

Generalized Covering Groups and Direct Limits

M. R. R. Moghaddam (Monatsh. Math. 90 (1980) 37-43.) showed that the Baer invariant commutes with the direct limit of a directed system of groups. In this paper, using the generalization of Schur's formula for the structure of a $\mathcal{V}$-covering group for a Schur-Baer variety $\mathcal{V}$, we show that the structure of a $\mathcal{V}$-covering group commutes with the direct limit of a directed system, in some senses. It has a useful application in order to extend some known structures of $\mathcal{V}$-covering groups for several famous products of finitely many to an arbitrary family of groups.

math.GR↗
arXiv2011

Some Baer Invariants of Free Nilpotent Groups

We present an explicit structure for the Baer invariant of a free $n$th nilpotent group (the $n$th nilpotent product of infinite cyclic groups, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*...\st{n}*\textbf{Z}$) with respect to the variety ${\cal V}$ with the set of words $V=\{[\ga_{c_1+1},\ga_{c_2+1}]\}$, for all $c_1\geq c_2$ and $2c_2-c_1>2n-2$. Also, an explicit formula for the polynilpotent multiplier of a free $n$th nilpotent group is given for any class row $(c_1,c_2,...,c_t)$, where $c_1\geq n$.

math.GR↗
arXiv2011

On Polynilpotent Covering Groups of a Polynilpotent Group

Let ${\cal N}_{c_1,...,c_t}$ be the variety of polynilpotent groups of class row $(c_1,...,c_t)$. In this paper, first, we show that a polynilpotent group $G$ of class row $(c_1,...,c_t)$ has no any ${\cal N}_{c_1,...,c_t,c_{t+1}}$-covering group if its Baer-invariant with respect to the variety ${\cal N}_{c_1,...,c_t,c_{t+1}}$ is nontrivial. As an immediate consequence, we can conclude that a solvable group $G$ of length $c$ with nontrivial solvable multiplier, ${\cal S}_nM(G)$, has no ${\cal S}_n$-covering group for all $n>c$, where ${\cal S}_n$ is the variety of solvable groups of length at most $n$. Second, we prove that if $G$ is a polynilpotent group of class row $(c_1,...,c_t,c_{t+1})$ such that ${\cal N}_{c'_1,...,c'_t,c'_{t+1}}M(G)\neq 1$, where $c'_i\geq c_i$ for all $1\leq i\leq t$ and $c'_{t+1}>c_{t+1}$, then $G$ has no any ${\cal N}_{c'_1,...,c'_t,c'_{t+1}}$-covering group. This is a vast generalization of the first author's result on nilpotent covering groups (Indian\ J.\ Pure\ Appl.\ Math.\ 29(7)\ 711-713,\ 1998).

math.GR↗
arXiv2011

Polynilpotent Multipliers of Finitely Generated Abelian Groups

In this paper, we present an explicit formula for the Baer invariant of a finitely generated abelian group with respect to the variety of polynilpotent groups of class row $(c_1,...,c_t)$, ${\cal N}_{c_1,...,c_t}$. In particular, one can obtain an explicit structure of the $\ell$-solvable multiplier (the Baer invariant with respect to the vaiety of solvable groups of length at most $\ell\geq 1,\ {\cal S}_{\ell}$.) of a finitely generated abelian group.

math.GR↗
arXiv2011

On Polynilpotent Multipliers of Free Nilpotent Groups

In this paper, we present an explicit structure for the Baer invariant of a free nilpotent group (the $n$-th nilpotent product of the infinite cyclic group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect to the variety of polynilpotent groups of class row $(c,1)$, ${\cal N}_{c,1}$, for all $c > 2n-2$. In particular, an explicit structure of the Baer invariant of a free abelian group with respect to the variety of metabelian groups will be presented.

math.GR↗
arXiv2011

The Baer Invariant of Semidirect and Verbal Wreath Products of Groups

W. Haebich (1977, Journal of Algebra {\bf 44}, 420-433) presented some formulas for the Schur multiplier of a semidirect product and also a verbal wreath product of two groups. The author (1997, Indag. Math., (N.S.), {\bf 8}({\bf 4}), 529-535) generalized a theorem of W. Haebich to the Baer invariant of a semidirect product of two groups with respect to the variety of nilpotent groups of class at most $c\geq 1,\ {\cal N}_c$. In this paper, first, it is shown that ${\cal V}M(B)$ and ${\cal V}M(A)$ are direct factors of ${\cal V}M(G)$, where $G=B\rhd<A$ is the semidirect product of a normal subgroup $A$ and a subgroup $B$ and $\cal V$ is an arbitrary variety. Second, it is proved that ${\cal N}_cM(B\rhd<A)$ has some homomorphic images of Haebich's type. Also some formulas of Haebich's type is given for ${\cal N}_cM(B\rhd<A)$, when $B$ and $A$ are cyclic groups. Third, we will present a formula for the Baer invariant of a $\cal V$-verbal wreath product of two groups with respect to the variety of nilpotent groups of class at most $c\geq 1$, where $\cal V$ is an arbitrary variety. Moreover, it is tried to improve this formula, when $G=A{\it Wr_V}B$ and $B$ is cyclic. Finally, a structure for the Baer invariant of a free wreath product with respect to ${\cal N}_c$ will be presented, specially for the free wreath product $A{\it Wr_*}B$ where $B$ is a cyclic group.

math.GR↗
arXiv2011

A Theorem of Van Kampen Type for Pseudo Peano Continuum Spaces

G. Conner and K. Eda (Topology and its Applications, 146, (2005), 317-328.) introduced a new construction of spaces from groups. They remarked that the construction is not categorical. In this paper, based on the work of Conner and Eda, we construct two new categories for which the functor of the fundamental group $π$ has a right adjoint and consequently is right exact and preseves direct limits. Also, we study the behavior of the functor $π$ on quotient spaces and give a new version of Van Kampen theorem for join of spaces in the new categories.

math.AT↗
arXiv2011

Topological Homotopy Groups

D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.

math.AT↗
arXiv2010

Some Inequalities for Nilpotent Multipliers of Powerful $p$-Groups

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 powerful p-group. Our results extend some of Lubotzky and Mann$^{,}$s (Journal of Algebra, 105 (1987), 484-505.) to nilpotent multipliers. Also, we give some explicit examples showing the tightness of our results and improvement some of the previous inequalities.

math.GR↗
arXiv2010

Some Properties of the Schur Multiplier with Algebraic Topological Approach

In this paper, using a relationship between the Schur multiplier of a group $G$, the fundamental group, and the second homology group of the Eilenberg-MacLane space of $G$, we present new proofs for some famous properties of the Schur multiplier and its structure for the free product and the direct product with an algebraic topological approach. Also, we try to find the Schur multiplier of a free amalgamated product in terms of Schur multipliers of its factors, which is a new result.

math.AT↗
arXiv2010

On the Order of Schur Multipliers of Finite Abelian p-Groups

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 $G$.

math.GR↗
arXiv2010

Some Inequalities for Nilpotent Multipliers of Finite Groups

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.

math.GR↗
arXiv2010

An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, for all $c_2\leq c_1\leq 2c_2$. As a consequence we determine necessary and sufficient conditions for such groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable. We also show that if $c_1\neq 1\neq c_2$, then a finitely generated abelian group is $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable if and only if it is capable. Finally we show that $\mathfrak{S}_2$-capability implies capability but there is a finitely generated abelian group which is capable but is not ${\mathfrak S}_2$-capable.

math.GR↗
arXiv2010

On Nilpotent Multipliers of Some Verbal Products of Groups

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 $n$, $c$.

math.GR↗
arXiv2010

Polynilpotent Capability of Finitely Generated Abelian Groups

In this paper we determine all finitely generated abelian groups which are varietal capable with respect to the variety of polynilpotent groups. This result is a vast generalization of the famous Baer's result about capability of finitely generated abelian groups.

math.GR↗
arXiv2010

On Topological Homotopy Groups of $n$-Hawaiian like spaces

By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, $\displaystyle{\varprojlim (Y_i^{(n)},y_i^*)}$, where $(Y_i^{(n)},y_i^*)=\bigvee_{j\leq i}(X_j^{(n)},x_j^*)$ is the wedge of $X_j^{(n)}$'s in which $X_j^{(n)}$'s are $(n-1)$-connected, locally $(n-1)$-connected, $n$-semilocally simply connected and compact CW spaces. In this paper, first we show that the natural homomorphism $\displaystyle{β_n:π_n(X,*)\rightarrow \varprojlim π_n(Y_i^{(n)},y_i^*)}$ is bijection. Second, using this fact we prove that the topological $n$-homotopy group of an $n$-Hawaiian like space, $π_n^{top}(X,x^*)$, is a topological group for all $n\geq 2$ which is a partial answer to the open question whether $π_n^{top}(X,x^*)$ is a topological group for any space $X$ and $n\geq 1$. Moreover, we show that $π_n^{top}(X,x^*)$ is metrizable.

math.AT↗