Non-abelian finite $p$-group in which every element has prime order?

Question

Let $G$ be a finite group such that for some prime $p$, every element has order $p$ . Can $G$ be non-abelian ?

My thoughts: Of-course $G$ is a $p$-group so has non-trivial center. If $G$ is non-abelian then $G$ has order $p^n$ where $n\ge 3$. For $n=3$ and $p=2$ , the only non-abelian groups are $D_4$ and $Q_8$ both of which has elements of order $4$, which doesn't satisfy my criteria. So we have to look beyond that.

Please help.

Answer 1

You forgot to mention nonidentity. You also want $p$ to be an odd prime, because if every nonidentity element has order $2$ the group is abelian.

And there is a nonabelian group of order $p^3$, the mod-$p$ Heisenberg group $$ \begin{bmatrix}1&*&*\\0&1&*\\0&0&1\end{bmatrix}\in SL(3,\mathbb{F}_p) $$ such that every nonidentity element has order $p$.