Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime.
Choose $\displaystyle x\in G$ so that $\displaystyle x\not = e$ which is possible by hypothesis. Construct the cyclic subgroup $\displaystyle \left< x\right>$. By hypothesis this must generate $\displaystyle G$ for this subgroup is nontrivial. Hence, $\displaystyle G$ is isomorphic to the cyclic group $\displaystyle \mathbb{Z}_n$. Now this group has proper non-trivial subgroups unless $\displaystyle n$ is a prime. Thus, $\displaystyle |G|=p$.