بیستمین سمینار جبر , 2009-04-22

#### Title : ( Some Notes On Polynilpotent Multipliers of p-Groups )

Authors: , Behrooz Mashayekhy Fard ,

Citation: BibTeX | EndNote

#### Abstract

In this article we show that if $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathcal{V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,...,c_s),\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ {\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathcal N}_{c_1,c_2,...,c_s}$, and $G={\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\bf {Z}}_{p^{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha_1}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\stackrel{n}{*}{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\bf {Z}}_{p^{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha_2}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\stackrel{n}{*}...\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\stackrel{n}{*}{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\bf{Z}}_{p^{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha t} }$ is the $n$th nilpotent product of some cyclic $p$-groups, where $c_1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\geq n$, $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha_1 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\geq \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha_2 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\geq...\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\geq \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha_t$ and $(q,p)=1$ for all prime $q$ less than or equal to $n$, then $|{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathcal N}_{c_1,c_2,...,c_s}M(G)|=p^{d_m}$ if and only if $G=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\underbrace{{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\bf {Z}}_{p}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\stackrel{n}{*}{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\bf {Z}}_{p}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\stackrel{n}{*}...\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\stackrel{n}{*}{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\bf{Z}}_{p }}_{m-copies}$ where $m=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\sum _{i=1}^t \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha_i$ and $d_m=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\chi_{c_s+1}...(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\chi_{c_2+1}(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\sum_{j=1}^n \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\chi_{c_1+j}(m)...)$.

#### Keywords

, Polynilpotent Multipliers; Nilpotent Products; Cyclic $p$, Groups.
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@inproceedings{paperid:1010671,
author = {, and Mashayekhy Fard, Behrooz},
title = {Some Notes On Polynilpotent Multipliers of p-Groups},
booktitle = {بیستمین سمینار جبر},
year = {2009},
location = {کرج, IRAN},
keywords = {Polynilpotent Multipliers; Nilpotent Products; Cyclic $p$-Groups.},
}