Exercise 5, § 2

Let Σ = (G, B, N, S) be a Tits system. Put T = B ∩ N and let Ċ be the normaliser of N.

a) Let b ∈ B ∩ Ċ. Show that bnb^{-1}n^{-1} ∈ B ∩ N for all n ∈ N (put bn = n'b and use Th. 1) and that b belongs to the intersection Ğ of the conjugates nBn^{-1} for n ∈ N. Show that Ğ ∩ N = T.

If Ğ = T, Σ is said to be saturated.

b) Put Ċ̃ = N.Ġ̃. Show that Ċ̃ is a subgroup of G containing Ġ̃ as a normal subgroup and that Ċ̃ ∩ B = Ġ̃. Show that the injection of N into Ċ̃ defines an isomorphism from the Weyl group W of Σ to Ċ̃/Ġ̃.

c) Show that (G, B, Ċ̃, j(S)) is a saturated Tits system, which is said to be associated to Σ.