We give a construction of Gusarov's groups [Gscr ]n of knots based on pure braid commutators, and show that any element of [Gscr ]n is represented by an infinite number of prime alternating knots of braid index less than or equal to n+1. We also study [Vscr ]n, the torsion-free part of [Gscr ]n, which is the group of equivalence classes of knots which cannot be distinguished by any rational Vassiliev invariant of order less than or equal to n. Generalizing the Gusarov–Ohyama definition of n-triviality, we give a characterization of the elements of the nth group of the lower central series of an arbitrary group.