In this post, we will classify groups of order pq, where p and q are primes with p<q. It turns out there are only two isomorphism classes of such groups, one being a cyclic group the other being a semidirect product.

Let be the group of order .

Case 1: does not divide .

By Sylow’s Third Theorem, we have , , , .

Since , or . Since and , we conclude . Similarly, since , or . Since , implies .

Let , be the Sylow -subgroup and Sylow -subgroup respectively. By Lagrange’s Theorem, . Thus . Since there is a non-identity element in which is not in . Its order has to be , thus is cyclic. Therefore .

Case 2: divides .

From previous arguments, hence is normal. Thus so is a subgroup of . thus . is cyclic, thus it has a unique subgroup of order , where .

Let and be generators for and respectively. Suppose the action of on by conjugation is , where . (We may conclude this since the action of on by conjugation is an automorphism which has order 1 or , thus it lies in .)

If , then .

If , then Choosing a different amounts to choosing a different generator for , and hence does not result in a new isomorphism class.