Classification of groups of order a product of two distinct primes
Suppose and are distinct prime numbers with . Then, there are two possibilities for the number of isomorphism classes of groups of order :
- If does not divide , then there is only one isomorphism class of groups of order , namely, the cyclic group.
- If divides , then there are two possibilities: the cyclic group of order and the semidirect product where is thought of as the additive group of integers mod and is identified with the subgroup of order in , which is cyclic of order .