An abelian p-group G is
thin if every map from a torsion complete group to G is small. The class of thin
groups is shown to be closed under arbitrary direct sums and extensions and hence
any direct sum of countable reduced p-groups is thin. An example shows that, unlike
for groups with no elements of infinite height, a reduced group may contain no
unbounded torsion complete subgroups and still fail to be thin. Finally,
these groups are used to settle questions in a certain relative homological
algebra.
