On Nilpotent Multipliers of Some Verbal Products of Groups

Abstract

The paper is devoted to finding a homomorphic image for the $c$-nilpotent multiplier of the verbal product of a family of groups with respect to a variety $\mathcal{V}$ when $\mathcal{V}\subseteq {\mathcal{N}}_c$ or ${\mathcal{N}}_c\subseteq \mathcal{V}$. Also a structure of the $c$-nilpotent multiplier of a special case of the verbal product, the nilpotent product, of cyclic groups is given. In fact, we present an explicit formula for the $c$-nilpotent multiplier of the $n$th nilpotent product of the group $G=\mathbf{Z}\stackrel{n}{\ast }...\stackrel{n}{\ast }\mathbf{Z}\stackrel{n}{\ast }{\mathbf{Z}}_{r_1}\stackrel{n}{\ast }...\stackrel{n}{\ast }{\mathbf{Z}}_{r_t}$, where $r_{i+1}$ divides $r_i$ for all $i$, $1\le i\le t-1$, and $(p,r_1)=1$ for any prime $p$ less than or equal to $n+c$, for all positive integers $n$, $c$.

16 pages