Abelian groups with a vanishing homology group

Abstract

An abelian group A is called absolutely abelian, if in every central extension N ↣ G ↠ A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H 2 A vanished. These groups, and more generally groups with H n A = 0 for some n, are characterized by elementary internal properties. (Here H ∗ A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2 ℵο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H 2 n A = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.