Given an Abelian group G and
a functorial characteristic subgroup hG, we study the extent to which G is
determined up to isomorphism by hG and GlhG. If G is a p-group or a mixed module
over a discrete valuation ring, we study the structure of G in terms of that of pλG
and G∕pλG, where λ is a limit ordinal. We also study a corresponding family of
subgroups of Abelian groups in general. For a countably generated reduced module
M of finite torsion-free rank over the ring Zp of integers localized at p, we obtain
necessary and sufficient conditions for M to be determined up to isomorphism by
pλM and M∕pλM.
