Exercise 1, § 2

Let G be a group, B and N two subgroups of G and S a subset of W = N/(B ∩ N). For all w ∈ W, put C(w) = BwB. Assume that conditions (T1) and (T2) of Def. 1 of no. 1 are satisfied, that whenever s ∈ S and w ∈ W at least one of the two relations C(s).C(w) = C(sw) and C(s).C(sw) = C(w) is satisfied, and that B ∪ C(s) is a subgroup of G for all s ∈ S. Show that condition (T3) is satisfied. Moreover, if the index of B in B ∪ C(s) is ≥ 3, then (G, B, N, S) is a Tits system.