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

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## Contents

## Statement

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 .

## Related facts

## Facts used

## Proof

### Direct proof using Sylow results

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

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