(B, N) pair

[1] BN pairs are closely related to reductive groups and the terminology in both subjects overlaps.

More precisely, the double cosets B\G/B are represented by a set of lifts of W to N.[3] Every parabolic subgroup equals its normalizer in G.[4] 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.

[5] More generally, this bijection extends to conjugacy classes of parabolic subgroups.

[6] 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.