Some outer commutator multipliers and capability of nilpotent products of cyclic groups

TL;DR

This paper derives an explicit formula for outer commutator multipliers of nilpotent products of cyclic groups within a specific variety and provides conditions for their capability, advancing understanding of their algebraic structure.

Contribution

It introduces a new explicit formula for outer commutator multipliers of nilpotent products of cyclic groups and establishes necessary conditions for their capability.

Findings

Explicit formula for outer commutator multipliers derived

Necessary conditions for groups to be capable established

Results applicable to specific varieties of nilpotent groups

Abstract

In this paper, first we obtain an explicit formula for an outer commutator multiplier of nilpotent products of cyclic groups with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, $\mathfrak{N}_{c}M(\mathbb{Z}\st{n}* \mathbb{Z}\st{n}* ... \st{n}* \mathbb{Z}\st{n}* \mathbb{Z}_{r_1}\st{n}* \mathbb{Z}_{r_2}\st{n}* ... \st{n}* \mathbb{Z}_{r_t})$ where $r_{i+1}\mid r_i \ \ (1\leq i\leq t-1)$, $c_1+c_2+1\geq n$, $2c_2-c_1>2n-2$ and $(p,r_1)=1$ for all prime less than or equal $c_1+c_2+n$, second we give a necessary condition for these groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable.