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This question already has an answer here:

If $|G|=p^{n}$

Then

Why is it that every subgroup of order $p^{n-1}$ is normal?

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marked as duplicate by Ben West, Alex M., Vlad, Shailesh, choco_addicted Apr 18 '16 at 8:12

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    $\begingroup$ Because it is a subgroup of index the smallest prime dividing the order of the group. Or because it cannot be self-normalizing since the group is nilpotent. $\endgroup$ – Tobias Kildetoft Apr 18 '16 at 7:32
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A subgroup of index $p$ where $p$ is the smallest prime dividing the order of the group is a normal subgroup.

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