Exercise 11, § 9

Let $V$ be a vector space of dimension 2 over a field $K$, and let a, b, c, d be four distinct points of the projective line $P(V)$. The cross-ratio of the quadruplet $(a, b, c, d)$ is called and denoted by

the set of elements $\xi \in K$ such that there exist two vectors $u, v$ in $V$ for which one has (with the notations of II, p. 133) $a = \pi(u)$, $b = \pi(v)$, $c = \pi(u + v)$, $d = \pi(u + \xi v)$. This definition extends immediately to every quadruplet of distinct points of a set endowed with a projective line structure (II, p. 138). (a) Now prove that 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 $P(V)$, and $\rho$ the set of conjugates of an element $\neq 1$ of $K^*$, there exists a point $d \in P(V)$ such that For d to be unique, it is necessary and sufficient that $\rho$ be reduced to a single element. (b) Now prove that one has and (where $\rho^{-1}$ (resp. $1 - \rho$) denotes 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 quadruplets of distinct points of $\mathbf{P}(V)$. For there to exist a bijective semilinear mapping of $V$ onto itself such that the bijective mapping $f$ of $\mathbf{P}(V)$ onto itself, obtained by passing to quotients, satisfies the conditions $f(a) = a', f(b) = b', f(c) = c', f(d) = d'$, it is necessary and sufficient that there exists an automorphism $\sigma$ of $K$ such that

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