International Journal of Mathematics, Game Theory and Algebra, Year (2006-3)

#### Title : ( On polynilpotent covering groups of a polynilpotent group )

Authors: Behrooz Mashayekhy Fard , - - ,

Citation: BibTeX | EndNote

#### Abstract

Let ${\\\\\\\\\\\\\\\\cal N}_{c_1,\\\\\\\\\\\\\\\\ldots,c_t}$ be the variety of polynilpotent groups of class row $(c_1,\\\\\\\\\\\\\\\\ldots,c_t)$. In this paper, first, we show that a polynilpotent group $G$ of class row $(c_1,\\\\\\\\\\\\\\\\ldots,c_t)$ has no any ${\\\\\\\\\\\\\\\\cal N}_{c_1,\\\\\\\\\\\\\\\\ldots,c_t,c_{t+1}}$-covering group if its Baer-invariant with respect to the variety ${\\\\\\\\\\\\\\\\cal N}_{c_1,\\\\\\\\\\\\\\\\ldots,c_t,c_{t+1}}$ 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 $(c_1,\\\\\\\\\\\\\\\\ldots,c_t,c_{t+1})$ such that ${\\\\\\\\\\\\\\\\cal N}_{c\\\\\\\\\\\\\\\'_1,\\\\\\\\\\\\\\\\ldots,c\\\\\\\\\\\\\\\'_t,c\\\\\\\\\\\\\\\'_{t+1}}M(G)\\\\\\\\\\\\\\\\neq 1$, 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 ${\\\\\\\\\\\\\\\\cal N}_{c\\\\\\\\\\\\\\\'_1,\\\\\\\\\\\\\\\\ldots,c\\\\\\\\\\\\\\\'_t,c\\\\\\\\\\\\\\\'_{t+1}}$-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).

#### Keywords

, $\\\\\\\\\\\\\\\\cal V$, covering group; polynilpotent groups; solvable, group.
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@article{paperid:1006584,
author = {Mashayekhy Fard, Behrooz and -, -},
title = {On polynilpotent covering groups of a polynilpotent group},
journal = {International Journal of Mathematics, Game Theory and Algebra},
year = {2006},
month = {March},
issn = {1060-9881},
keywords = {$\\\\\\\\\\\\\\\\cal V$-covering group; polynilpotent groups; solvable-group.},
}