In a group of order $m p^n$ for $p$ prime, if $k<n$, is there an element of order $p^k$?

Question

Let $G$ a group of order $mp^n$ where $p$ is prime. Let $k\leq n$. Is there an element of order $p^k$ ?

---

Since $p$ divide $|G|$, by Cauchy theorem, there is $g\in G$ s.t. $g$ has order $p$. I can't do better. I tried to use the fact that there is a $p-$Sylow group, but it just confirm the fact that there is an element of order $p$, not of order $p^k$.

Answer 1

No, for instance when $G=\mathbb{F}_p^3$, every element has order $p$.