Find a group $G$ that contains elements $a$ and $b$ such that $a^2=e$, $b^2=e$, but the order of the element $ab$ is infinite.
Clearly $G$ cannot be abelian. So I looked at two commonly known non-abelian groups, namely
(ii) 2 by 2 matrices
Neither of these seem to work. Any help would be much appreciated, guys.