Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G is cyclic and |G|=p^2 where p is prime.
Note that once you construct it cannot be and cannot be . Since that is a subgroup and it has no other proper non-trivial subgroups it follows that . Since the group is cyclic it is isomorphic to for some . By the properties of cyclic groups we can show that only are the ones with this unique non-trivial proper subgroup property.