Q:

If G is not commutative

Edit: Since I did not provide enough detail in my explanation in OP: I have tried to prove this for the general case, but have not come across a suitable proof. I was unsure if I then needed to prove this generally, of if an example was sufficient. If there is a general proof for this, it would be appreciated. 
Let $G$ be a group and let $o(a)=2, o(b)=3$, for some $a,b \in G$. Show that, if $G$ is not commutative, then $o(ab)$ is not necessarily $6$.

A:

In $\;S_3\;$ , take $\;a=(12)\;,\;\;b=(123)\;$ . We don't even need to check what the product is to know it can't have order six.

