I have looked at many proofs that show that there exists a normal subgroup of size $p^{k-1}$, but no proofs that all subgroups of size $p^{k-1}$ are normal.
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2Look up the general theorem that if $p$ is the smallest prime dividing the order of a finite group $G$, all subgroups of $G$ with index $p$ are normal. That explains what you are asking and why a subgroup of index $2$ is always normal. – KCd Mar 08 '21 at 22:21
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1What have you tried? This is a pretty normal group theory problem. – Rushabh Mehta Mar 08 '21 at 22:21
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1Its index is the smallest prime that divides the order of the group. Then apply this – Arturo Magidin Mar 08 '21 at 22:21
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2@ArturoMagidin Wild that you remember your answers from more than 8 years ago. – Rushabh Mehta Mar 08 '21 at 22:23
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2@DonThousand: That particular problem comes up a lot, so I know I've answered it. Then I searched for it. – Arturo Magidin Mar 08 '21 at 22:23
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1@DonThousand nice use of "normal" in your comment. – KCd Mar 08 '21 at 22:24
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2@KCd I thought it was pretty punny :) – Rushabh Mehta Mar 08 '21 at 22:36