Problem: Prove the following: Every Abelian group $G$ with $|G| = pq$ where $p$ and $q$ are two distinct primes, is cyclic.

Attempt: According to the first Sylow theorem, exists a Sylow $p$-subgroup of $G$. Let $n_p(G)$ be amount of Sylow $p$-subgroups of $G$. Then by the third Sylow theorem we have $n_p(G) \equiv 1 \mod p$ and $n_p(G)$ divides $q$. From this it follows that $n_p(G) = 1$ or $n_p(G) = q$.

Now I wanted to treat the two cases separately.

Case 1: Let $n_p(G) = 1$. Let $P$ be the unique Sylow $p$-subgroup. Then $P$ is a normal subgroup of $G$ because $P$ is conjugated to itself. Hence $$ [G: P] = \frac{ |G|}{|P|} = \frac{pq}{ |P|} = q. $$ This is because $|P| = p$, since a finite group is a $p$-group if and only if its order is a power of $p$. The above then shows that $G/P$ is cyclic since $q$ is prime.

Now I wanted to construct an isomorphism of some kind, which relates $G$ to $G/P$ and then conclude that since $G/P$ is cyclic, so is $G$. But I can't find the right map.

Case2: For this case I didn't have much inspiration. Suppose $n_p(G) = q$. I know there are then $q$ amount of Sylow $p$-subgroups of $G$, which are conjugated to each other. How to continue?

Help is appreciated!

  • 4
    $\begingroup$ This is very complicated: $G$ has an element $a$ of order $p$ and an element $b$ of order $q$. Consider the element $ab$. Prove that it generates $G$. $\endgroup$ Sep 1, 2018 at 16:22
  • $\begingroup$ Close to being a duplicate of this more general variant. As I happened to answer that version, I won't cast the first vote to close as a dupe (it would be a decisive one due to my dupehammer privilege in abstract-algebra). There may also equally good other dupe targets. $\endgroup$ Sep 2, 2018 at 6:40

1 Answer 1


The suggestion in the comment is perfectly good. To gain some perspective, I would like to point out the Fundamental Theorem of Abelian groups for you:


So your group is isomorphic to $(\mathbb{Z}_p, +) \times (\mathbb{Z}_q, +)$.

There is another general theorem you should know (and usually all these important facts are explained in any introductory course before Sylow's theorems): the direct product of two nontrivial groups $G,H$ is cyclic if and only if the groups $G,H$ are finite cyclic groups with coprime order.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.