# Why is every subgroup of order $p^{n-1}$ normal? [duplicate]

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

Then

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

• 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. – Tobias Kildetoft Apr 18 '16 at 7:32

A subgroup of index $p$ where $p$ is the smallest prime dividing the order of the group is a normal subgroup.