TITLE: Order of product of two elements in an abelian group
QUESTION [15 upvotes]: Let $G$ be an abelian group. Let $a, b \in G $ and let their order be $m$ and $n$ be respectively. Is it always true that order of $ab$ is $lcm(m,n)$?
What if $m$ and $n$ are coprime to each other?

REPLY [20 votes]: You can prove that "anything" could happen for the order of a product.
In more precise words:
For any integers $m;n;r > 1$, there exists a finite group $G$ with elements $a$ and $b$ such that $a$ has order $m$, $b$ has order $n$, and $ab$ has order $r$.
For a beautiful proof, see Theorem 1.64 in http://www.jmilne.org/math/CourseNotes/GT.pdf