Exercise 12, § 1

Let (W, S) be a Coxeter system such that m(s, s') is even if s ≠ s' and let X be a subset of S. Show that there exists a unique homomorphism f_X from W to W_X such that f_X(s) = s for s ∈ X and f_X(s) = 1 for s ∈ S - X. Deduce that W is the semi-direct product of W_X and the kernel of f_X. Show that if X ⊂ Y ⊂ S, there exists a unique homomorphism f_{X,Y} from W_Y to W_X such that f_X = F_{X,Y} ∘ f_Y and that W can be identified with a subgroup of the projective system thus obtained from the $W_X$ as $X$ runs through the filtered set of finite subsets of $S$.