# Classification of groups of order a product of two distinct primes

## Statement

Suppose $p$ and $q$ are distinct prime numbers with $p < q$. Then, there are two possibilities for the number of isomorphism classes of groups of order $pq$:

1. If $p$ does not divide $q - 1$, then there is only one isomorphism class of groups of order $pq$, namely, the cyclic group.
2. If $p$ divides $q - 1$, then there are two possibilities: the cyclic group of order $pq$ and the semidirect product $\mathbb{Z}_q \rtimes \mathbb{Z}_p$ where $\mathbb{Z}_q$ is thought of as the additive group of integers mod $q$ and $\mathbb{Z}_p$ is identified with the subgroup of order $p$ in $\mathbb{Z}_q^\ast$, which is cyclic of order $q - 1$.

## Facts used

1. Every Sylow subgroup is cyclic implies metacyclic

## Proof

### Direct proof using Sylow results

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### Proof using fact (1) (strong thing to use in proof)

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