Question

Prove that abelian group of order pq (p,q distinct prime) is cyclic.

Answer 1

Question 1. Prove that abelian group of order $pq$ ($p,q$ are distinct primes) is cyclic.

Solution. Let $G$ be an abelian group of order $pq$. By the fundamental theorem of finite abelian groups we have two cases: either $G \cong \mathbb{Z}_{pq}$ (the cyclic group of order $pq$), or $G \cong \mathbb{Z}_p \oplus \mathbb{Z}_q$ (the direct sum of cyclic groups of orders $p$ and $q$). Prove that $Z_p \oplus \mathbb{Z}_q \cong \mathbb{Z}_{pq}$. Indeed, let $a$ generate $\mathbb{Z}_p$, i.e. $pa = 0$ in $\mathbb{Z}_p$, and $b$ generate $\mathbb{Z}_q$, i.e. $qb = 0$ in $\mathbb{Z}_q$. Then consider the pair $(a,b) \in \mathbb{Z}_p \oplus \mathbb{Z}_q$. Note that

$$pq(a,b) = (pqa,pqb) = (q(pa),p(qb)) = (q \cdot 0, p \cdot 0) = (0, 0),$$

therefore, the order of $(a,b)$ divides $pq$. Since $(a,b) \neq (0,0)$, then the order cannot be equal to 1. It cannot be also equal to $p$ or $q$, because $qa \neq 0$ and $pb \neq 0$ (for example, $qa = 0$ would imply that $p$ divides $q$, which is impossible, since $p$ and $q$ are distinct primes). Therefore, the order of $(a,b)$ is $pq$ and it is the order of $\mathbb{Z}_p \oplus \mathbb{Z}_q$. Thus, $\mathbb{Z}_p \oplus \mathbb{Z}_q$ is the cyclic group, generated by $(a,b)$.