Possible order of $ab$ when orders of $a$ and $b$ are known.
Solution 1:
Let $a$ and $b$ be elements of a group $G$. If $a$ has order $m$ and $b$ has order $n$, what can we say about the order of $ab$? The following theorem, proved in Milne's lecture notes on group theory, shows that we can say nothing at all.
THEOREM 1.64 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 $r$.