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A Magnus-Witt type isomorphism for non-free groups. (English) Zbl 1048.20017
Let \(\gamma_nG\) denote the lower central series of the group \(G\) and \(\Gamma G=\bigoplus_{n\geq 1}\gamma_nG/\gamma_{n+1}G\) be the Lie algebra of quotients of the \(\gamma_nG\). Let \(L(G_{ab})\) be the free Lie algebra on the Abelian group \(G_{ab}=G/\gamma_2G\). The author proves that certain Baer invariants of \(G\) are torsion when \(G\) has torsion second integral homology. This result is used to show that if \(G\) has torsion-free Abelianisation then \(\Gamma G\) is isomorphic to \(L(G_{ab})\).

MSC:
20F14 Derived series, central series, and generalizations for groups
20F40 Associated Lie structures for groups
20J05 Homological methods in group theory
18G50 Nonabelian homological algebra (category-theoretic aspects)
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