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If p and q are prime, show every proper subgroup of a group of order pq 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 pq must have order 1, p, q, or pq. If the subgroup were order 1 or pq, it would be either <e> or the whole group, both trivial. So, any nontrivial subgroup would have order p or q, both prime. Since all groups of prime order are cyclic, it must be true that all proper subgroups of a group of order pq are cyclic.

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