Journal of Advanced Research in Pure Mathematics, Volume (1), No (1), Year (2009-1) , Pages (39-48)

Title : ( On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic p-Groups )

Authors: Behrooz Mashayekhy Fard , Fahimeh Mohammadzadeh ,

Citation: BibTeX | EndNote

Abstract

In this article we show that if {cal V} is the variety of polynilpotent groups of class row (c_1,c_2,...,c_s), {mathcal N}_{c_1,c_2,...,c_s}, and G cong{bf {Z}}_{p^{alpha_1}}stackrel{n}{*}{bf {Z}}_{p^{plpha_2}}stackrel{n}{*}...stackrel{n}{*}{bf{Z}}_{p^{alpha_t} } is the nth nilpotent product of some cyclic p-groups, where c_1geq n, alpha_1geq alpha_2 geq...geq alpha_t and (q,p)=1 for all primes q less than or equal to n, then |{mathcal N}_{c_1,c_2,...,c_s}M(G)|=p^{d_m} if and only if Gcong{bf {Z}}_{p}stackrel{n}{*}{bf {Z}}_{p}stackrel{n}{*}...stackrel{n}{*}{bf{Z}}_{p } (m-copies), where m=sum _{i=1}^t alpha_i and d_m=chi_{c_s+1}(...(chi_{c_2+1}(sum_{j=1}^n chi_{c_1+j}(m)))...)$. Also, we extend the result to the multiple nilpotent product Gcong{bf {Z}}_{p^{alpha_1}}stackrel{n_1}{*}{bf {Z}}_{p^{alpha_2}}stackrel{n_2}{*}...stackrel{n_{t-1}}{*}{bf{Z}}_{p^{alpha_t} }, where c_1geq n_1geq...geq n_{t-1}. Finally a similar result is given for the {Z}}_{p^{alpha_1}}stackrel{n}{*}{bf {Z}}_{p^{alpha_2}}stackrel{n}{*}...stackrel{n}{*}{bf{Z}}_{p^{alpha_t}} with the different conditions n geq c and (q,p)=1 for all primes q less than or equal to n+c.

Keywords

, Polynilpotent multiplier; Nilpotent product; Cyclic group; Finite p , group; Elementary Abelian p , group.
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@article{paperid:1010930,
author = {Mashayekhy Fard, Behrooz and Fahimeh Mohammadzadeh},
title = {On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic p-Groups},
journal = {Journal of Advanced Research in Pure Mathematics},
year = {2009},
volume = {1},
number = {1},
month = {January},
issn = {1943-2380},
pages = {39--48},
numpages = {9},
keywords = {Polynilpotent multiplier; Nilpotent product; Cyclic group; Finite p -group; Elementary Abelian p -group.},
}

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%0 Journal Article
%T On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic p-Groups
%A Mashayekhy Fard, Behrooz
%A Fahimeh Mohammadzadeh
%J Journal of Advanced Research in Pure Mathematics
%@ 1943-2380
%D 2009

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