TITLE: Possible order of $ab$ when orders of $a$ and $b$ are known.
QUESTION [10 upvotes]: Let $a,b\in G$ be elements of a finite group $G$. We know $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. In dependence of $m$ and $n$ what are the possible values of $\operatorname{ord}(ab)$?
So strictly speaking I'm looking for a function $f:\mathbb N^2\to\mathcal{P}(\mathbb N)$.
Note that the restriction of $G$ being finite is no weakening of the problem. In case $n,m>1$ we could always add $\infty$ to the result of $f(m,n)$.

REPLY [14 votes]: 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 $r$.
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