Polynilpotent Multipliers of Some Nilpotent Products of Cyclic Groups

TL;DR

This paper derives explicit formulas for polynilpotent multipliers of certain nilpotent products of cyclic groups, expanding understanding of their structure in the context of nilpotent and polynilpotent varieties.

Contribution

It provides the first explicit formulas for the c-th nilpotent multiplier of specific nilpotent products of cyclic groups and their polynilpotent multipliers.

Findings

Explicit formula for the c-th nilpotent multiplier of the given group class

Calculation of polynilpotent multiplier with respect to specified polynilpotent variety

Conditions under which the formulas are valid, such as divisibility and coprimality

Abstract

In this article, we present an explicit formula for the $c$th nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c\geq 1$) of the $n$th nilpotent product of some cyclic groups $G={\mathbb {Z}}\stackrel{n}{*} ... \stackrel{n}{*}{\mathbb {Z}}\stackrel{n}{*} {\mathbb {Z}}_{r_1}\stackrel{n}{*} ... \stackrel{n}{*}{\mathbb{Z}}_{r_t}$, (m-copies of $\mathbb {Z}$), where $r_{i+1} | r_i$ for $1 \leq i \leq t-1$ and $c \geq n$ 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 ${\mathcal N}_{c_1,c_2,...,c_t}$, where $c_1 \geq n.$