Exercise 23, § 1

We retain the notation of Exerc. 16. Let $\mathrm{Aut}(A)$ be the group of automorphisms of the apartment $A$. Show that if $\varphi \in \mathrm{Aut}(A)$, then $\varphi$ permutes the walls of $A$, and $\varphi t \varphi^{-1} \in T$ for all $t \in T$ (use Exerc. 18). Deduce that $W$ (identified with a subgroup of $\mathrm{Aut}(A)$) is normal and that $\mathrm{Aut}(A)$ is the semi-direct product of the subgroup $E$ of automorphisms preserving the chamber $C$ by $W$. Show that the action of $\mathrm{Aut}(A)$ on $W$ defines an isomorphism from $E$ to the group of automorphisms of the Coxeter system $(W, S)$, or to that of the Coxeter graph of $(W, S)$ (cf. also Exerc. 19).