Exercise 9, § 1

Let (W, S) be a Coxeter system with matrix (m(s, s')), and let W^+ be the subgroup of W consisting of the elements of even length. Let s_0 ∈ S. Put g_s = ss_0. Show that the family (g_s)_{s ∈ S - {s_0}} and the relations g_s^{m(s, s_0)} = 1 for m(s, s_0) ≠ ∞ and (g_s g_{s'}^{-1})^{m(s, s')} = 1 for s, s' ∈ S - {s_0} and m(s, s') ≠ ∞, form a presentation of W^+. (Let H^+ be the group defined by the above presentation. Show that there exists an automorphism α of H^+, with square the identity, that transforms g_s to g_s^{-1} for all s ∈ S - {s_0}. If H_α is the semi-direct product of {1, -1} and H^+, relative to α, define mutually inverse homomorphisms H_α → W and W → H_α.) Show that if the elements of S are conjugate (cf. Prop. 3), the group W^+ is the commutator subgroup of W (remark that the elements g_s are then commutators).