I was trying to prove that a group $G$ of order $p^n$, where $p$ is prime, has a normal subgroup of each order $p^i$, $1\leq i\leq n$. This has been asked before (for example here) but I want to know if my aproach is correct.
We proceed by induction on $|G|$. The case $n=1$ is clear. Now, suppose a group of order $p^{n-1}$ has a normal subgroup of any possible order and let $|G|=p^n$. Take $x$ of order $p$ in the center of $G$. Consider the group $G/\langle x\rangle$ of order $p^{n-1}$. Now, by hypothesis $G/\langle x\rangle$ has a normal subgroup $G/H_i$ of order $p^i$ for each $1\leq i<n$ where $\langle x\rangle \leq H_i\leq G$ by Correspondence Theorem. Now, $H_i\trianglelefteq G$ and $|H_i|=p^{n-i}$.
