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 V when V ⊆ N c or N c ⊆ 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 nth nilpotent product of the group G = Z ∗ n ⋯ ∗ n Z ∗ n Z r 1 ∗ n ⋯ ∗ n Z r t , where r i + 1 divides r i for all i, 1 ⩽ i ⩽ t − 1 , and ( p , r 1 ) = 1 for any prime p less than or equal to n + c , for all positive integers n, c.