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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?

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    Have you prove that $G/H$ verifies the induction hypothesis?2011-11-13
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    Well $G/H$ has size $p^n$, which is finite and of a size the hypothesis applies to, and it is abelian because $G$ is assumed to be abelian. So it seems to satisfy the induction hypothesis.2011-11-13
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    One part of the induction hypothesis is that it has a unique subgroup of order $p$.2011-11-13
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    @Iasafro I've added an argument for that, see the edit above.2011-11-13
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    You have made a circular argument. You cannot conlude that $G/H$ has a unique subgroup of order $p$ by applying that it is cyclic, i.e. that has a unique generator $gH$, since $G/H$ being cyclic is a consecuence of the induction hypothesis, that is, of having $G/H$ a unique subgroup of order $p$. In brief, being cyclic has to be a consecuence of $G/H$ having a unique subgroup of order $p$ and not in the other sense.2011-11-13
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    @Iasafro Indeed :( Perhaps induction is not the best way, or I haven't applied it correctly. I'll settle for any method of proof at this point.2011-11-13

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