Show that $\sim$ is an equivalence relation on $G\setminus \{e\}$ if and only if $o(g)$ is prime for each $g \in G\setminus \{e\}$,
where $\sim$ is defined on $G\setminus \{e\}$ by $g \sim h$ if and only if $g^k = h$ for some $k \geq 1$,
$g, h \in G\setminus \{e\}$, $e$ is the identity element of $G$, and $o(g)$ is the order of $g$.
It's pretty easy to prove reflexivity and transitivity, but I can't seem to be able to put a finger on symmetricity. That's also where the prime number condition seems to come in.
Hope it's an interesting problem. Any help greatly appreciated.