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This is the problem:

Let $p$ be a prime and $G$ a $p$-group. Prove that $Z(G)$ (center of $G$) is cyclic if and only if $G$ has a unique normal subgroup of order $p$.

I can't see why having a unique subgroup implies a cyclic center and vice-versa.

In my notes about $p$-groups the only results says that $Z(G) \neq \{e\}$ if $G$ is $p$-group.

Thanks!

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closed as off-topic by Derek Holt, Saad, Namaste, Chris Custer, Xander Henderson Apr 10 '18 at 0:29

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  • $\begingroup$ It would help if you told us how much you know about this stuff. For example, in a $p$-group, a minimal normal group must be contained in the center. This already brings you very close. $\endgroup$ – verret Apr 9 '18 at 22:16