Exercise 3, § 2

With the notation and assumptions of no. 3, let $u$ be an automorphism of the affine space $E$ that permutes the hyperplanes $L_{\alpha, k}$ ($\alpha \in R, k \in \mathbf{Z}$). Show that $u$ is a displacement. (If $u_0$ is the linear map associated to $u$, show that the transpose of $u$ leaves $R$ stable, and hence belongs to the group $A(R)$.) Deduce that $G$ is the normaliser of $W_a$ in the group of automorphisms of the affine space $E$.