Suppose that G is the general linear group GLn(K) over a field K. We take B to be the upper triangular matrices, T to be the diagonal matrices, and N to be the monomial matrices, i.e. matrices with exactly one non-zero element in each row and column. There are n − 1 generators, represented by the matrices obtained by swapping two adjacent rows of a diagonal matrix. The Weyl group is the symmetric group on n letters. More generally, if G is a reductive group over a field K then the group G=G(K) has a BN pair in which B=P(K), where P is a minimal parabolic subgroup of G, and N=N(K), where N is the normalizer of a split maximal torus contained in P. In particular, any finite group of Lie type has the structure of a BN-pair. Over the field of two elements, the Cartan subgroup is trivial in this example. A semisimple simply-connected algebraic group over a local field has a BN-pair where B is an Iwahori subgroup. Properties Bruhat decomposition The Bruhat decomposition states that G = BWB. More precisely, the double cosets B\G/B are represented by a set of lifts of W to N. Parabolic subgroups Every parabolic subgroup equals its normalizer in G. Every standard parabolic is of the form BW(X)B for some subset X of S, where W(X) denotes the Coxeter subgroup generated by X. Moreover, two standard parabolics are conjugate if and only if their sets X are the same. Hence there is a bijection between subsets of S and standard parabolics. More generally, this bijection extends to conjugacy classes of parabolic subgroups. Tits's simplicity theorem BN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if G has a BN-pair such that B is a solvable group, the intersection of all conjugates of B is trivial, and the set of generators of W cannot be decomposed into two non-empty commuting sets, then G is simple whenever it is a perfect group. In practice all of these conditions except for G being perfect are easy to check. Checking that G is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showing that a group is perfect is usually far easier than showing it is simple. Citations References Section 6.2.6 discusses