Polynilpotent Multipliers of Some Nilpotent Products of Cyclic Groups

Abstract

In this article, we present an explicit formula for the cth nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most c ≥ 1) of the nth nilpotent product of some cyclic groups \({G = {\mathbb{Z}}\stackrel{n}{*}\cdots \stackrel{n}{*}{\mathbb{Z}}\stackrel{n}{*} {\mathbb {Z}}_{r_1}\stackrel{n}{*}\cdots\stackrel{n}{*}{\mathbb{Z}}_{r_t}}\), (m-copies of \({\mathbb {Z}}\)), where r i+1 | r i for 1 ≤ i ≤ t − 1 and c ≥ n such that (p, r 1) = 1 for all primes p less than or equal to n. Also, we compute the polynilpotent multiplier of the group G with respect to the polynilpotent variety \({{\mathcal N}_{c_1,c_2,\ldots ,c_t}}\), where c 1 ≥ n.