Subgroups of finite index in a procyclic group

Question

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 im 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

Answer 1

I think there's only one procyclic group of any order $p^n$ (just like cyclic groups): http://books.google.com.au/books?id=47ouE_XSJZYC&pg=PA51&lpg=PA51&dq=procyclic+group&source=bl&ots=xJ_S1YJGYj&sig=da4qaWYBPDlkxBTcIaJcCEm8Clo&hl=en&sa=X&ei=wOt7UKmlDq_BiQfm04CYCA&ved=0CDcQ6AEwAw