Proposition 4

Let $P$ be a subgroup of $G$.

a) $P$ is parabolic if and only if there exists a subset $X$ of $S$ such that $P$ is conjugate to $G_X$ (cf. no. 5 for the definition of $G_X$).

b) Let $X, X' \subset S$ and $g, g' \in G$ be such that $P = g G_X g^{-1} = g' G_{X'} {g'}^{-1}$. Then, $X = X'$ and $g' g^{-1} \in P$.

Assertion a) follows from Th. 3, b). Under the hypotheses of b), we have

and Prop. 3 shows that $g^{-1} g' \in G_X$. Hence, $G_{X'} = G_X$ and $X' = X$ by Th. 3, b). Finally, which proves b).

If the parabolic subgroup $P$ is conjugate to $G_X$, where $X \subset S$, then $P$ is said to be of type X.