On Nilpotent Multipliers of Some Verbal Products of Groups

TL;DR

This paper investigates the structure and explicit formulas for the $c$-nilpotent multipliers of verbal and nilpotent products of groups, providing new insights into their homomorphic images and algebraic properties.

Contribution

It derives explicit formulas and structural descriptions for the $c$-nilpotent multiplier of verbal products, especially nilpotent products of cyclic groups, under specific divisibility and coprimality conditions.

Findings

Explicit formula for the $c$-nilpotent multiplier of the $n$th nilpotent product of certain cyclic groups.

Structural description of the $c$-nilpotent multiplier for verbal products within a variety ${ m extbf{N}}_c$.

Identification of conditions under which the homomorphic image of the $c$-nilpotent multiplier can be characterized.

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 $G= {\bf {Z}}\stackrel{n}{*}...\stackrel{n}{*}{\bf {Z}}\stackrel{n}{*} {\bf {Z}}_{r_1}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{r_t}$, where $r_{i+1}$ divides $r_i$ for all $i$, $1 \leq i \leq t-1$, and $(p,r_1)=1$ for any prime $p$ less than or equal to $n+c$, for all positive integers $n$, $c$.