  We prove that groups that are mod-p-homology equivalent are isomorphic modulo
any term of their derived p-series, in precise analogy to Stallings' 1963
result for the lower-central p-series. Similarly spaces that are mod-p-homology
equivalent have fundamental groups that are isomorphic modulo any term of their
p-derived series. Various authors have related the ranks of the successive
quotients of the lower central p-series and of the derived p-series of the
fundamental group of a 3-manifold M to the volume of M, to whether certain
subgroups of the fundamental group of M are free, whether finite index
subgroups of the fundamental group of M map onto non-abelian free groups, and
to whether finite covers of M are ``large'' in various other senses.
  Specifically, let A be a finitely-generated group and B be a finitely
presented group. If a homomorphism induces an isomorphism (respectively
monomorphism) on H_1(- ;Z_p) and an epimorphism on H_2(- ;Z_p), then for each
finite n, it induces an isomorphism (respectively monomorphism) between the
quotients of A and B by the n-th terms of their respective p-derived series. In
fact we prove a stronger version that is analogous to Dwyer's extension of
Stallings' theorem.
