[Math] Any subgroup of index $p$ in a $p$-group is normal.

Question

Let $p$ be a prime number and $G$ a finite group where $|G|=p^n$, $n \in \mathbb{Z_+}$. Show that any subgroup of index $p$ in it is normal in $G$. Conclude that any group of order $p^2$ have a normal subgroup of order $p$, but without using the Sylow theorems.

Answer 1

I think another approach in light of Don's answer can be:

Lemma: Suppose $G$ is a $p$-group and $H < G$, then $H\lneqq N_G(H)$.

Here we know that $[G:H]=p$ then $H$ is a proper subgroup of $G$. So the lemma tells us in this group we have $H$ as a proper subgroup of its normalizer in $G$. In fact our conditions make $N_G(H)$ be $G$ itself and this means that $H\vartriangleleft G$.