Polynilpotent Multipliers of some Nilpotent Products of Cyclic Groups II

TL;DR

This paper derives explicit formulas for the polynilpotent multipliers of certain nilpotent products of cyclic groups, extending understanding of their structure within specific varieties and conditions.

Contribution

It provides new explicit formulas for the $c$th polynilpotent multiplier of complex nilpotent products of cyclic groups under specific divisibility and coprimality conditions.

Findings

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

Calculation of the polynilpotent multiplier with respect to the ${ m N}_{c_1,c_2,...,c_s}$ variety.

Conditions under which the formulas hold, including divisibility and coprimality constraints.

Abstract

This article is devoted to present an explicit formula for the $c$th nilpotent multiplier of nilpotent products of some cyclic groups $G={\bf {Z}}\stackrel{n_1}{*}{\bf {Z}}\stackrel{n_2}{*}...\stackrel{n_{t-1}}{*}{\bf {Z}}\stackrel{n_{t}}{*} {\bf {Z}}_{m_{t+1}}\stackrel{n_{t+1}}{*}{\bf {Z}}_{m_{t+2}}\stackrel{n_{t+2}}{*}...\stackrel{n_{k}}{*}{\bf{Z}}_{m_{k+1}}$, where $m_{i+1} | m_i$ for all $t+1 \leq i \leq k$ and $c \geq n_1\geq n_2\geq ...\geq n_t\geq ...\geq n_{k}$ such that $ (p,m_{t+1})=1$ for all prime $p \leq n_1$. Moreover, we compute the polynilpotent multiplier of the group $G$ with respect to the polynilpotent variety ${\mathcal N}_{c_1,c_2,...,c_s}$, where $c_1 \geq n_1.$