Let $\displaystyle G$ be a finite group of order $\displaystyle 4n+2$ for some integer $\displaystyle n$. Let $\displaystyle g_1, g_2 \in G$ be such that $\displaystyle o(g_1)\equiv o(g_2) \equiv 1 \, (\mbox{mod} 2)$. Show that $\displaystyle o(g_1g_2)$ is also odd.

I found a solution to this recently but I think that solution uses a very indirect approach. Not saying that that solution was bad.. just indirect. So I wanted to see a more direct proof. I will post the solution I am talking about later in this thread once this is solved since if I post it now it might interfere with the thought process. I am sorry but I have no ideas of my own on how to go about doing it in a different way. Please help.