Order of product of two elements in an abelian group

Solution 1:

When the group is finite and Abelian, you can show that if $d=\text{ord}(ab)$ where $m=\text{ord}(a)$ and $n=\text{ord}(b)$ then $$d|\frac{mn}{\gcd(m,n)}=\text{lcm}(m,n)$$ and $$\frac{mn}{\gcd(m,n)^2}|d.$$ ${}{}{}{}{}$ In particular this means that if $m$ and $n$ are coprime the order is multiplicative.

Solution 2:

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