Exercise 18, § 9

Let $E = \mathbf{P}(V)$ be a projective plane over a field $K$, $u$ a bijective mapping of $E$ onto itself, obtained by passing to quotients from a bijective semilinear mapping $v$ of $V$ onto itself, relative to an automorphism $\sigma$ of $K$. a) Prove that the following four properties are equivalent: $\alpha)$ for every $x \in E$, $x$, $u(x)$ and $u^2(x)$ are on a straight line; $\beta)$ every right of $E$ contains a point invariant under $u$; $\gamma)$ through every point of $E$ passes a right invariant under $u$; $\delta)$ for every right $D$ of $E$, the three rights $D$, $u(D)$ and $u^2(D)$ have a common point. (Prove first that $\alpha$) and $\gamma)$ are equivalent; deduce, by duality (II, p. 203, Exercise 12), that $\beta)$ and $\delta)$ are equivalent; prove finally that $\gamma)$ implies $\beta)$ and deduce by duality that $\beta)$ implies $\gamma)$.) b) Suppose that $u$ has the properties stated in a). Prove that if there exists in $E$ a right $D$ invariant under $u$ and containing only one point $a$ invariant under $u$, $u$ is obtained by passing to quotients from a transvection $v$ of $V$ (II, p. 206, Exercise 11). (Prove that every right invariant under $u$ contains $a$; by considering a right not passing through $a$, prove that there exists a right $D_0$ passing through $a$ and containing at least two points invariant under $u$; conclude that all points of $D_0$ are necessarily invariant under $u$.) c) Suppose that $u$ has the properties stated in a). Prove that if there exists in $E$ a right $D$ invariant under $u$ and containing only two points invariant under $u$, $u$ is obtained by passing to quotients from a dilation $v$ of $V$ (II, p. 206, Exercise 11). (If $a$, $b$ are the two points of $D$ invariant under $u$, prove that every right invariant under $u$ passes through $a$ or through $b$; then observe that there exist at least two other points $c$, $d$ distinct from $a$, $b$ and invariant under $u$, and consequently that the right $D_{cd}$ passes through $a$ or $b$; conclude by proving that all points of $D_{cd}$ are invariant under $u$.) d) Suppose that $u$ has the properties stated in a) and that every right of $E$ invariant under $u$ contains at least three distinct points invariant under $u$; there then exists in $E$ a projective frame (II, p. 202, Exercise 10) each point of which is invariant under $u$; conclude that there exists a basis $(e_i)_{1 \leq i \leq 3}$ of $V$ such that $u$ is obtained by passing to quotients from a semilinear mapping $v$ of $V$ onto itself such that $v(e_i) = e_i$ for $1 \leq i \leq 3$. The set of points of $E$ invariant under $u$ is then the projective plane $\mathbf{P}(V_0)$, where $V_0$ is the vector space over the field $K_0$ of the invariants of $\sigma$, generated by $e_1$, $e_2$, $e_3$. e) Suppose henceforth that $u$ satisfies the conditions of a) and d), and that neither $u$ nor $u^2$ is the identity. Prove that there exists $\gamma \in K$ such that $\gamma^\sigma = \gamma$ and

for every $\xi \in K$ (use the condition $\alpha$) of a), and the existence of $\zeta \in K$ such that $\zeta^\sigma \neq \zeta$). Prove that $\gamma \neq -1$, then that, for every $\xi \in K$ such that $\xi^\sigma \neq \xi$, one has

(applying (1) by replacing $\xi$ by $\xi^2$); extend (2) to every $\xi \in K$ by noting that $\xi = \eta - \zeta$, where $\eta^\sigma \neq \eta$ and $\xi^\sigma \neq \zeta$. Conclude that $\gamma \neq 1$, then deduce from (1) and (2) that one has

for every $\xi \in K$, and that $\gamma + \gamma^{-1}$ belongs to the center of $K$. Converse. Give an example where $\gamma$ does not belong to the center of $K$, but where $\gamma + \gamma^{-1}$ is in this center.