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17.85 (2010)

Open

Let $\mathfrak{V}$ be a variety of groups such that any free group in $\mathfrak{V}$ has torsion-free integral homology groups in all dimensions. Is it true that $\mathfrak{V}$ is abelian?

Contributor: R. Mikhailov

Progress

It is known that free metabelian groups may have 2-torsion in the second homology groups, and free nilpotent groups of class 2 may have 3-torsion in the fourth homology groups.

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