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 [𝔑 _{c 1} , 𝔑 _{c 2} ], for all c ₂ ≤ c ₁ ≤ 2c ₂. As a consequence, we determine necessary and sufficient conditions for such groups to be [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable. We also show that if c ₁ ≠ 1 ≠ c ₂, then a finitely generated abelian group is [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable if and only if it is capable. Finally, we show that 𝔖₂-capability implies capability, but there is a capable finitely generated abelian group which is not 𝔖₂-capable.

Key Words:

• Baer invariant
• Finitely generated abelian group
• Outer commutator variety
• Varietal capability

2000 Mathematics Subject Classification:

• 20E34
• 20E10
• 20F12
• 20F18

ACKNOWLEDGMENT

The first author was in part supported by a grant from IPM No. 85200018.

Notes

Communicated by A. Olshanskii.