p-group has normal subgroup of any prime power order

Proposition

A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k\le n$.

\begin{proof} By induction hypothesis, apply Cauchy theorem to $Z(G/N_k)$ where $N_k$ is normal subgroup of order $p^k$. \end{proof}