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I have been reading about procyclic groups, in particular $\hat{\mathbb{Z}}$, and in many places they claim that if $H$ is a subgroup of a procyclic group with finite index then it is unique, i.e if any other subgroup has the same index then it must be in fact $H$. But I am not quite sure how to prove this, so I wanted to see if I could get some hints on how to do this.

Thank you.

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