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.