A finite abelian group whose order is the product of two distinct primes is cyclic

Prove that if G is an abelian group of order pq, where p and q are distinct primes, then G is cyclic.


By Cauchy’s Theorem, there exist elements x, y \in G of order p and q, respectively. Since G is abelian, |xy| = \mathsf{lcm}(p,q) = pq. Thus \langle xy \rangle = G, hence G is cyclic.

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