Exercise 8, § 2

Let $(G, B, N, S)$ be a Tits system; put $T = B \cap N$ and $W = N/T$. Let $\tilde{G}$ be a group containing G as a normal subgroup. Assume that for all $h \in \tilde{G}$, there exists $g \in G$ such that $hBh^{-1} = gBg^{-1}$ and $hNh^{-1} = gNg^{-1}$. Let $\hat{B}$ (resp. $\hat{N}$) be the normaliser of B (resp. N) in $\tilde{G}$; put $\Gamma = \hat{B} \cap \hat{N}$, $\tilde{N} = \Gamma N$ and $\tilde{T} = \tilde{N} \cap B$.

a) Show that $\tilde{G} = \Gamma.G$, $\hat{B} = \Gamma.B$, $\Gamma \cap B = \Gamma \cap G$ and $\tilde{T} = (\Gamma \cap B).T$. The groups $\Omega = \Gamma/(\Gamma \cap B)$, $\tilde{G}/G$ and $\hat{B}/B$ are thus canonically isomorphic. If $\Phi \subset \Omega$ and if H is a subgroup of G containing $\Gamma \cap B$, we denote by $\Phi H$ the union of the subsets $\varphi H$ for $\varphi \in \Phi$.

b) Show that $\tilde{T}$ is normal in $\tilde{N}$ (to show that $n \gamma n^{-1} \in \tilde{N}$ for $n \in \tilde{N}$ and $\gamma \in \Gamma \cap B$, use Exerc. 5 a)), that $N \cap \tilde{T} = T$ and that $\Gamma \cap \tilde{T} = \Gamma \cap B$. The injection of N (resp. $\Gamma$) into $\tilde{N}$ thus allows W (resp. $\Omega$) to be identified with a subgroup of $\tilde{W} = \tilde{N}/\tilde{T}$. Show that $\Omega$ normalises S and that $\tilde{W}$ is the semi-direct product of $\Omega$ and W.

c) Show that, for all $s \in S$ and all $u \in \tilde{W}$,

d) Show that the map $u \mapsto BuB$ is a bijection from $\tilde{W}$ to $B \setminus \tilde{G}/B$ (use Th. 1 and the fact that $\Gamma$ normalises B).

e) Let $\mathscr{B}$ be the set of pairs $(\Phi, X)$, where $\Phi$ is a subgroup of $\Omega$ and X is a subset of X normalised by $\Phi$. Put $G_{(\Phi, X)} = \Phi G_X = B \Phi W_X B$ (with the notation of no. 5). Show that the map $(\Phi, X) \mapsto G_{(\Phi, X)}$ is a bijection from $\mathscr{B}$ to the set of subgroups of $\tilde{G}$ containing B. Generalize assertions b) and c) of Th. 3 and Prop. 2 of no. 5.

Show that the normaliser of $G_{(\Phi, X)}$ in $\tilde{G}$ is the subgroup $G_{(\Phi', X)}$, where $\Phi'$ is the set of elements of $\Omega$ normalizing both $\Phi$ and X.

f) Show that $G_{(\Phi, X)}$ is a maximal subgroup of $\tilde{G}$ if and only if one of the following two conditions is satisfied:

(i) $X = S$ and $\Phi$ is a maximal subgroup of $\Omega$;

(ii) $\Phi = \Omega$ and $\Phi$ acts transitively on $S - X$ (which is non-empty).

Show that $G_{(\Phi, X)}$ is a maximal subgroup in the set of subgroups of $\tilde{G}$ not containing $G$ if and only if

(iii) $X \neq S$, $\Phi$ is the normaliser of $X$ in $\Omega$ and acts transitively on $S - X$.

g) Let $\Phi$ be a normal subgroup of $\Omega$ and put $G' = \Phi G$, $B' = \Phi B$, $N' = \Phi N$ and $T' = B' \cap N'$. Show that $T' = \Phi \tilde{T}$ and that $T'$ is normal in $N'$ if and only if every element of $\Phi$ commutes with every element of $W$. Show that $G'$ is then normal in $\tilde{G}$, that the injection of $N$ into $N'$ defines an isomorphism $j$ from $W$ to $W' = N'/T'$ and that $(G', B', N', S')$ (with $S' = j(S)$) is a Tits system.