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Throughout this abstract, G is a topological Abelian group and is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism given by hh | D is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these.1. There are (many) nonmetrizable, noncompact, determined groups.2. If the dense subgroup Di determines Gi with Gi compact, then ⊕iDi determines ΠiGi. In particular, if each Gi is compact then ⊕iGi determines ΠiGi.3. Let G be a locally bounded group and let G+ denote G with its Bohr topology. Then G is determined if and only if G+ is determined.4. Let non (𝒩) be the least cardinal κ such that some X ⊂ 𝕋 of cardinality κ has positive outer measure. No compact G with w(G) ≥ non (𝒩) is determined; thus if non (𝒩) = N1 (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω.Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is κ = non (𝒩)? κ = N1?

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