This seems to be a very easy exercise and yet I fail at the last part:
I want to show that the rotations around a point $O$ are an Abelian sub-group of the group of all movements in a plane which fulfils Hilbert's combination-, order- and congruence-axioms. I have already shown that for rotations $\varphi$ and $\psi$, $\varphi \circ \psi$ is a rotation and that the inverse of a rotation is a rotation as well. So what I have left to do is to show that $\varphi \circ \psi = \psi \circ \varphi$.
I tried writing $\varphi = \tau_1 \circ \sigma$ and $\psi = \tau_2 \circ \sigma$ for three reflections $\tau_1, \tau_2$ and $\sigma$ but this did not yet lead me to any success. What is the trick to show commutativity?
Thanks a lot for any answers.