Let $A$ and $B$ be two rotations of the Euclidean plane $\mathbb{E}^{2}$ about centres $a$ and $b$ respectively. Show that the conjugate $A B A^{-1}$ is also a rotation and find its fixed point. When do $A$ and $B$ commute? Show that the commutator $A B A^{-1} B^{-1}$ is a translation.

Deduce that any group of orientation-preserving isometries of the Euclidean plane either fixes a point or is infinite.