An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

Details An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

2010-12-15

arxiv.org/abs/1012.3244v1

Communications in Algebra?, 38: 588-600, 2010

Mathematics

Group Theory

19 pages

Scientific paper

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.

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