## 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.