showing a set is not a subgroup Let $G$ be the orthogonal subgroup $O_2$. Show that the set $\{g \in G : g^2= e\}$ is not a subgroup of $G$
The question before says let $G$ be an abelian group and I can see where I have used that fact. It lets us write $a^2b^2=(ab)^2$ and so $ab \in G$. But I cant find a way to 'get out' of $G$. 

Somewhere in your proof you have used that $(gh)^2=g^2h^2.$ You need $G$ to be abelian for this. 
In general $(gh)^2=ghgh$ and you cannot swap the inner $h$ and $g$ unless $G$ is abelian.
