My goal is to prove this:
If $G$ is a finite abelian $p$-group with a unique subgroup of size $p$, then $G$ is cyclic.
I tried to prove this by induction on $n$, where $|G| = p^n$ but was not able to get very far with it at all (look at the edit history of this post to see the dead ends). Does anyone have any ideas for a reasonably elementary proof of this theorem?