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Cyclic groups have at most one subgroup of any given finite index. Can we describe the class of all groups having such property?

Thank you!

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    So, the natural question seems to be: 'Is it true that for any such group G, the canonical residually finite quotient of G (ie the quotient of G by the normal subgroup of elements that are contained in every finite-index subgroup) is cyclic?' Does anyone have a counterexample?2010-11-27
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    Actually, perhaps Arturo's answer proves exactly that. Arturo?2010-11-27
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    On further thought, I think so. I posted this as an answer.2010-11-28
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    OK, I now conjecture that arbitrary (ie possibly infinitely generated) $G$ has this property if and only if its profinite completion is a profinite cyclic group. I haven't had time to think about a proof, but I doubt it's hard.2010-12-06

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