Let k be an algebraically closed field of characteristic p ≥ 0. We will take the numbering of Dynkin diagrams of irreducible root systems as given in Table 9.1.

Exercise 20.1 (Existence of graph automorphisms)

1. (a) Show how to reduce the proof of Theorem 11.12 on the existence of graph automorphisms to the case of simple groups of simply connected type.

2. (b) Verify the details of the proof for type SLn, n ≥ 3.

3. (c) Show that a suitable element of GO2n induces a non-trivial graph automorphism of SO2n, n ≥ 2.

[Hint: For (c) consider the element given in Example 22.9(2).]

Exercise 20.2 Let G be a group with a BN-pair, with W = N/(B ∩ N) generated by a set of involutions S. For w ∈ W write ℓ(w) for the length of a shortest expression w = s1 … sr with si ∈ S. Show the following:

1. (a) If s ∈ S, w ∈ W with ℓ(ws) ≥ ℓ(w) then BẇB · BṡB ⊆ BẇsB.

2. (b) If s ∈ S, w ∈ W with ℓ(ws) ≤ ℓ(w) then BẇB · BṡB has non-empty intersection with BẇB.

3. (c) If ℓ(ws) < ℓ(w), then ṡ ∈ Bẇ-1BẇB.