We provide some characterizations of precompact abelian groups G whose dual group Gp∧\documentclass[12pt]{minimal}
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\begin{document}$$G_p^\wedge $$\end{document} endowed with the pointwise convergence topology on elements of G contains a nontrivial convergent sequence. In the special case of precompact abelian torsion groups G, we characterize the existence of a nontrivial convergent sequence in Gp∧\documentclass[12pt]{minimal}
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\begin{document}$$G_p^\wedge $$\end{document} by the following property of G: No infinite Hausdorff quotient group of G is countable. Also, we present an example of a dense subgroup G of the compact metrizable group Z(2)ω\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}(2)^\omega $$\end{document} such that G is of the first category in itself, has measure zero, but the dual group Gp∧\documentclass[12pt]{minimal}
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\begin{document}$$G_p^\wedge $$\end{document} does not contain infinite compact subsets. This complements a result of J.E. Hart and K. Kunen (2005) on convergent sequences in dual groups. Making use of the group G we construct the first example of a precompact Pontryagin reflexive abelian group which is of the first Baire category.
