Among compact Hausdorff groups whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. We also prove that the abstract commutator subgroup is closed for every closed normal subgroup of .
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Dan Segal, Nikolay Nikolov, On normal subgroups of compact groups. J. Eur. Math. Soc. 16 (2014), no. 3, pp. 597–618DOI 10.4171/JEMS/441