Example 1

Let k be a field, n an integer \geqslant 0, G = \mathbf{GL}(n, k), and let (G, B, N, S) be the Tits system described in no. 2. Let U be the strictly upper triangular subgroup of G, i.e. the subgroup of B consisting of the matrices whose diagonal entries are equal to 1. Condition (1) in Th. 5 is immediate, and so is (2) since U is soluble. Condition (4) is satisfied if $n \geq 2$. One can show (cf. Algebra, Chap. II, § 10, Exerc. 13) that (3) is satisfied if $n \geq 3$ and Card$(k) \geq 4$. Under these conditions, we conclude that $G_1 / (G_1 \cap Z)$ is simple and that $G_1 \cap Z$ is the centre of $G_1$ (cf. Remark 2).

When $k$ is commutative, $G_1 = \mathbf{SL}(n, k)$ (cf. Algebra, Chap. III, § 8, no. 9).