Exercise 2, § 2

Let G be a group, B and N two subgroups of G and S a subset of W = N/(B ∩ N). Let Z be a normal subgroup of G contained in B. Let B' and N' be the canonical images of B and N in G' = G/Z. Show that the canonical map from N to N' defines an isomorphism from W to W' = N'/(B' ∩ N'). Let S' be the image of S under this isomorphism. Show that (G', B', N', S') is a Tits system if and only if (G, B, N, S) is one.