An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

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 $c_2\le c_1\le 2c_2$. 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 $c_1\ne 1\ne c_2$, 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.

19 pages