  We provide some characterizations of precompact abelian groups $G$ whose dual
group $G_p^\wedge$ endowed with the pointwise convergence topology on elements
of $G$ contains a nontrivial convergent sequence. In the special case of
precompact abelian \emph{torsion} groups $G$, we characterize the existence of
a nontrivial convergent sequence in $G_p^\wedge$ by the following property of
$G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present
an example of a dense subgroup $G$ of the compact metrizable group
$\mathbb{Z}(2)^\omega$ such that $G$ is of the first category in itself, has
measure zero, but the dual group $G_p^\wedge$ does not contain infinite compact
subsets. This complements Theorem 1.6 in [J.E.~Hart and K.~Kunen, Limits in
function spaces and compact groups, \textit{Topol. Appl.} \textbf{151} (2005),
157--168]. As a consequence, we obtain an example of a precompact reflexive
abelian group which is of the first Baire category.
