Let $R$ be any fixed rotation and $F$ be any fixed reflection in a dihedral group. Prove that $(FR)(FR)=e$

Question

Let $R$ be any fixed rotation and $F$ be any fixed reflection in a dihedral group. Prove that $(FR)(FR)=e$

I saw this assumption in the back of my textbook for a solution and I don't know how to prove it. It's true of all the little drawings of squares and triangles I make, but how is this mathematically proven?

Why isn't $(FR)^{-1}=R^{-1}F$ as the socks-shoes property would suggest?

Answer 1

This holds because a flip followed by a rotation (or vice versa) is, again, a flip, so has order two. Don't be afraid to use your geometric intuition here!

And what about $(FR)^{-1}=R^{-1}F^{-1}$? It just means $R^{-1}F^{-1}=FR$ in this context.