[Math] Subgroups of order $p$ and $p^{n-1}$ in a group of order $p^n$.

Question

I have a group $G$ of order $p^n$ for $n \ge 1$ and $p$ a prime. I am looking for two specific subgroups within $G$: one of order $p$ and one of order $p^{n-1}$. I don't think I would use the Sylow theorems here because those seem to apply to groups with a "messier" order than simply $p^n$. Would Cauchy's Theorem allow me to generate the two requisite subgroups? I could use it to find an element of order $p$ and an element of order $p^{n-1}$ and then consider the cyclic subgroups generated by these two elements?

Answer 1

Hints:

1) Use the class formula and deduce that $\,|Z(G)|>1\,$

2) Use induction now to show that $\,G\,$ has a normal subgroup of order $\,p^k\,\,,\,\,\forall\,\,0\le k\le n\,$