Let $G$ be a finite group and $g,h \in G$ both have order 2. Determine the possible orders of $gh$.
So I first think of this in terms of symmetric groups. Obviously $g$ and $h$ could be the same transposition in $S_n$ and thus $gh$ is the identity (order 1). They could also be two disjoint transpositions and have order 2. And if they are two non-disjoint transpositions they would have order 3.
For order 4 or higher I couldn't really construct more examples, but I think that is mainly because I am limiting myself to the symmetric groups. Perhaps there are finite groups that are more complex that would help me arrive at an answer faster... Any thoughts or hints?