Let denote, respectively, the number of sylow-p subgroups and sylow-q subgroups of G.

From Sylow's Theorems, for some integer is a factor of pq. Now, since p and q are (co)prime and p<q, , and that one sylow-q subgroup is normal in G.

Also, since PQ = G and .

Now, since p divides q-1, has a unique subgroup P' such that .

So, we let a and b be generators for P and Q, respectively. Suppose the action of a on Q by conjugation is in . Then

Since a different amounts to choosing a different generator a for P, it does not result in a new isomorphism class. THEREFORE, there are exactly two isomorphism classes of groups of order pq.

Does this help?