[algebra] A little problem (ii)

I'm following the great course on abstract algebra offered online by Harvard, lectured by Benedict Gross. So far, i'm having a great time. I'm trying to do the exercises (Algebra, Artin, 1st ed. 1991), to gain a better understanding and intuition on the matter, so i'll be posting exercises that i find interesting. Mind you, i'm just getting started, so interesting may be a bit too far-fetched...

Exercise (2.2.20.a): Let $G$ be an abelian group, and $a,b \in G$. Let $m,n$ be the orders of the groups generated by $\langle a \rangle$ and $\langle b \rangle$. What can you say about the order of $\langle ab \rangle$?

We consider the elements generated by $\langle ab \rangle$:

• 1 (identity).
• $ab$.
• $(ab)^2 = abab$. Since the group is commutative, we can reorder like this: $aabb = a^2b^2$.
• $(ab)^k = a^kb^k$, more generally.

Therefore, once we reach either $(ab)^m$ or $(ab)^n$, whichever's the smallest of $m$ and $n$, we will be left with the rest of the cyclic group generated by $a$ or $b$. So, we can conclude that the order of $\langle ab \rangle$ is in fact the order of $a$ or $b$.