  Let P be the direct product of countably many copies of the additive group Z
of integers. We study, from a set-theoretic point of view, those subgroups of P
for which all homomorphisms to Z annihilate all but finitely many of the
standard unit vectors. Specifically, we relate the smallest possible size of
such a subgroup to several of the standard cardinal characteristics of the
continuum. We also study some related properties and cardinals, both
group-theoretic and set-theoretic. One of the set-theoretic properties and the
associated cardinal are combinatorially natural, independently of any
connection with algebra.
