We define a new infinite sequence of invariants, d&d1;n for n ≥ 0, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. We show that they give lower bounds for the Thurston norm. Moreover, we show that they give better estimates for the Thurston norm than the previously known bounds given by the Alexander norm, d&d1;0 . To do this, we exhibit 3-manifolds whose Alexander norm is trivial but whose d&d1;n are strictly increasing and can be made arbitrarily large. Other applications are made to detecting 3-manifolds that fiber over S 1 and to detecting 4-manifolds that admit no symplectic structure.