How to prove: Projective transformation composition of at most three involutions?

Question

How do you show that any projective transformation of $\mathbb RP^1$ is the composition of at most three involutions of $\mathbb RP^1$?

Thanks

Answer 1

Let $ f_1, f_2$ and $f_3$ be involutions and $f\colon \mathbb{R}\mathrm{P}^1 \to \mathbb{R}\mathrm{P}^1 $ be a projective transformation. Now let $A, B, C$ and $D$ be four points in $\mathbb{R}\mathrm{P}^1$. Weobtain the following construction: $$ P \overset{f_1}{\to} Q \overset{f_2}{\to} R \overset{f_3}{\to} S \\ Q \to P \to S \to R \\ R \to S \to P \to Q \\ S \to R \to Q \to P $$ Now define $f_A := f_1, f_B := f_2 \circ f_1$ and $f_C := f_3 \circ f_2 \circ f_1$. These are projective transformations.

Since there are only three ways to interchange four letters such that we have involution for any change, so $f$ can't be the composition of four or more projective involutions.