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I have the following question: Prove that a finite $p-$group $G$ with a unique subgroup $H$ of index $p$ is cyclic.

Since the subgroup is characteristic it is normal. I showed that either the center of the group $Z(G)$ is a subgroup of $H$ or $G=Z(G)H$ and the index $Z(G)/(Z(G)\cap H)$ is $p$. But I couldn't continue from this. Any help would be great.

delueze
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1 Answers1

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The answer is due to Alexander Gruber (see here): If $G$ is finite then every proper subgroup is contained in a maximal subgroup, so every proper subgroup is contained in $H$, the unique subgroup of index $p$. Therefore we have $\langle x \rangle = G$ for any $x\in G\setminus H$.

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Dietrich Burde
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  • Why is any maximal subgroup of $G$ actually contained in $H$ ? Is there any simple argument for this (e.g. without Sylow theorems, as it is done in Marshall Hall's book, theorem 4.3.2) ? – Alphonse Nov 07 '16 at 20:52