On Nilpotent Multipliers of Some Verbal Products of Groups
Azam Hokmabadi, Behrooz Mashayekhy

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 -nilpotent multiplier of the verbal product of a family of groups with respect to a variety when or . Also a structure of the -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 -nilpotent multiplier of the th nilpotent product of the group , where divides for all , , and for any prime less than or equal to , for all positive integers , .
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