2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128757Recently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain homology localization of groups. In order to give a concrete combinatorial description and existence proof of the necessary homology localization, we introduce a new version of algebraic closures of groups with coefficients by considering a certain type of equations.28 pages; to appear in Proceedings of the London Mathematical SocietyGeometric TopologyGroup TheoryPrimary 20J05, 57M07, Secondary 55P60, 57M27text