On Polynilpotent Covering Groups of a Polynilpotent Group

Abstract: 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).