Let $G$ be an abelian group of order $pq$ with $gcd(p,q)=1$. Show that $G$ is cyclic.

Can someone help me finish the proof? I am basing off this answer.

Let $G$ be an abelian group of order $pq$ with $gcd(p,q)=1$. If there exist elements $a$ and $b$ such that $|a|=p$ and $|b|=q$, show that $G$ is cyclic.

After showing that $g^{pq}=e$, what do I do for the cases where $g=e$, $g^{p}=e$, and $g^{q}=e$? Any hints?

Solution 1:

Now $g=ab$.
Case 1: $g=e$. Then $a=b^{-1}$. Hence $p=|a|=|b^{-1}|=|b|=q$, a contradiction.
Case 2: $g^p=e$. That is, $$(ab)^p=e\implies a^pb^p=e\implies b^p=e$$ which means that $q$ divides $p$, a contradiction.
Case 3: $g^q=e$. Similar to case 2.