I'm trying to prove the following:
G is a group with order $\ge 2$ with no proper, non-trivial subgroups. G must be finite of prime order.
Consider $g \neq e \in G$ (we can do this since order of $G$ is at least 2). Since $G$ has no proper, non-trivial subgroups, $<g>$ can't be a proper subgroup of $G$. Since it clearly can't be $e$, we must have $<g> = G$.
I'm not sure why it has to be finite though...
Thanks guys, Mariogs