Exercise 7, § 2

Let G be a group acting on a set E. Then G is said to act doubly-transitively on E if, whenever $x, y, x', y' \in E$ are such that $x \neq y$ and $x' \neq y'$, there exists an element $g \in G$ such that $g.x = x'$ and $g.y = y'$.

a) Let $(G, B, N, S)$ be a Tits system whose Weyl group W is of order 2. Show that G acts doubly-transitively on $G/B$.

b) Let G be a group acting doubly transitively on a set E. Assume that Card(E) $\geqslant 3$. Let $e \in E$ and let B be the stabilizer of e. Let $x \in E$, with $x \neq e$, and let $n \in G$ be such that $n(e) = x$ and $n(x) = e$. Let N be the subgroup of G generated by n. Let s be the canonical image of n in N/T. Show that $(G, B, N, \{s\})$ is a Tits system whose Weyl group is of order 2.