Classification of groups of order a product of two distinct primes

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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.

Related facts

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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