  In 1964, John Stallings established an important relationship between the
low-dimensional homology of a group and its lower central series. We establish
a similar relationship between the low-dimensional homology of a group and its
derived series. We also define a torsion-free-solvable completion of a group
that is analogous to the Malcev completion, with the role of the lower central
series replaced by the derived series. We prove that the torsion-free-solvable
completion is invariant under rational homology equivalence.
