If and are prime, show every proper subgroup of a group of order is cyclic.
By Lagrange's Theorem, the order of a subgroup must divide the order of the group. Thus, any subgroup of a group of order must have order , , , or . If the subgroup were order or , it would be either or the whole group, both trivial. So, any nontrivial subgroup would have order or , both prime. Since all groups of prime order are cyclic, it must be true that all proper subgroups of a group of order are cyclic.