For a precompact abelian torsion group G, a nontrivial convergent sequence exists in the pointwise dual G_p^ exactly when G has a countably infinite Hausdorff quotient.
Arhangel’skii, M
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On convergent sequences in dual groups
For a precompact abelian torsion group G, a nontrivial convergent sequence exists in the pointwise dual G_p^ exactly when G has a countably infinite Hausdorff quotient.