An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

TL;DR

This paper explicitly describes the Baer invariant of finitely generated abelian groups relative to certain varieties and establishes conditions for their capability, revealing relationships between different capability notions.

Contribution

It provides an explicit structure for the Baer invariant with respect to $[rak{N}_{c_1}, rak{N}_{c_2}]$ and characterizes capability conditions for finitely generated abelian groups.

Findings

Determines necessary and sufficient conditions for $[rak{N}_{c_1}, rak{N}_{c_2}]$-capability.

Shows equivalence of capability and $[rak{N}_{c_1}, rak{N}_{c_2}]$-capability when $c_1,c_2 eq 1$.

Identifies that $rak{S}_2$-capability implies capability, but not vice versa.

Abstract

We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, for all $c_2\leq c_1\leq 2c_2$. As a consequence we determine necessary and sufficient conditions for such groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable. We also show that if $c_1\neq 1\neq c_2$, then a finitely generated abelian group is $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable if and only if it is capable. Finally we show that $\mathfrak{S}_2$-capability implies capability but there is a finitely generated abelian group which is capable but is not ${\mathfrak S}_2$-capable.