On Polynilpotent Covering Groups of a Polynilpotent Group

Abstract

Let ${\mathcal{N}}_{c_1,...,c_t}$ 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 ${\mathcal{N}}_{c_1,...,c_t,c_{t+1}}$-covering group if its Baer-invariant with respect to the variety ${\mathcal{N}}_{c_1,...,c_t,c_{t+1}}$ is nontrivial. As an immediate consequence, we can conclude that a solvable group $G$ of length $c$ with nontrivial solvable multiplier, ${\mathcal{S}}_{n}M(G)$, has no ${\mathcal{S}}_n$-covering group for all $n>c$, where ${\mathcal{S}}_n$ 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 ${\mathcal{N}}_{c_1^{\prime },...,c_t^{\prime },c_{t+1}^{\prime }}M(G)\ne 1$, where $c_i^{\prime }\ge c_i$ for all $1\le i\le t$ and $c_{t+1}^{\prime }>c_{t+1}$, then $G$ has no any ${\mathcal{N}}_{c_1^{\prime },...,c_t^{\prime },c_{t+1}^{\prime }}$-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).

10 pages

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