I am searching for an example of a non-abelian group $G$, s.t. all elements in $G$ have order $p$, where $p$ is a prime number. For $p=2$ such an example does not exist.
How did I come up with such a question? If $G$ is abelian with the property above, we can define a $\mathbb{F}_p$-vector space structure on $G$. And my proof used that $G$ is a $\mathbb{Z}$-module. So the above question arose naturally.
An example is the following group, for an odd prime $p>2$:
$$G= \langle x, y, z | x^p=1, y^p=1, z^p=1, [x,z]=1, [y,z]=1, [x,y]=z^{-1} \rangle$$ is non-abelian of order $p^3$, and all its non-trivial elements have order $p$. The group is exactly the Heisenberg group over $\mathbb{F}_p$.
More generally, this question is related to the restricted Burnside problem:
Proposition: A finite group $G$ has the property that all non-trivial elements have the same order $p$ if and only if $p$ is prime and $G\ne 1$ is a quotient of $B_0(m,p)$ for some $m.$
