Exercise 11, § 9

Let $V$ be a 2-dimensional vector space over a field $K$ and $a, b, c, d$ four distinct points of the projective line $\mathbf{P}(V)$. The cross-ratio of the quadruple $(a, b, c, d)$, denoted by $\begin{bmatrix} a & b \\ d & c \end{bmatrix}$, is the set of elements $\xi \in K$ such that there exist two vectors $u, v$ in $V$ for which (in the notation of no. 6) $a = \pi(u), b = \pi(v), c = \pi(u + v), d = \pi(u + \xi v)$. This definition extends immediately to any quadruple of distinct points of a set with a projective line structure (no. 11).

(a) Show that $\begin{bmatrix} a & b \\ d & c \end{bmatrix}$ is the set of conjugates of an element $\neq 1$ of the multiplicative group $K^*$ and conversely that, when $a, b, c$, are distinct points of $\mathbf{P}(V)$ and $\rho$ the set of conjugates of an element $\neq 1$ of $K^*$, there exists a point $d \in \mathbf{P}(V)$ such that $\begin{bmatrix} a & b \\ d & c \end{bmatrix} = \rho$. For $d$ to be unique, it is necessary and sufficient that $\rho$ consist of a single point.

(b) Show that

and (denoting by $\rho^{-1}$ (resp. $1 - \rho$) the set of conjugates $\lambda \xi^{-1} \lambda^{-1} = (\lambda \xi \lambda^{-1})^{-1}$ (resp. $1 - \lambda \xi \lambda^{-1} = \lambda (1 - \xi) \lambda^{-1}$), where $\xi$ is an element of $\rho$).

(c) Let $(a, b, c, d), (a', b', c', d')$ be two quadruples of distinct points of $\mathbf{P}(V)$. For there to exist a bijective semi-linear mapping of $V$ onto itself such that the bijective mapping $f$ of $\mathbf{P}(V)$ onto itself, obtained when passing to the quotients, satisfies the conditions $f(a) = a', f(b) = b', f(c) = c', f(d) = d'$, it is necessary and sufficient that there exist an automorphism $\sigma$ of $K$ such that

For there to exist a transformation $f$ of the projective group $\mathbf{PGL}(V)$ satisfying the above conditions, it is necessary and sufficient that