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If $G$ is abelian, $|G|=p^k$ for some $k\in\mathbb{Z}$, and $G$ has a unique subgroup of order $p$, then $G$ is cyclic.

I'm having difficulty getting anywhere on this problem. The supposition that there is a unique subgroup leads me to believe Sylow's theorem should be used.

In particular, if there is a unique Sylow p-subgroup P, then $P\triangleleft G$. So for all $a\in P, g\in G$, $gag^{-1}\in P$. Moreover, by Cauchy's theorem, $P$ is cyclic.

How can I relate these facts to show that $G$ is cyclic?

  • Do you know the fundamental theorem of finitely generated abelian groups? – Ethan Alwaise Nov 28 '16 at 06:35
  • They haven't said anything about any Sylow $p$-subgroups, but rather a subgroup of order $p$. Also, the group is abelian, so any subgroup is automatically normal. – Arthur Nov 28 '16 at 06:36

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