In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems. Definition A (B, N) pair is a pair of subgroups B and N of a group G such that the following axioms hold: G is generated by B and N. The intersection, T, of B and N is a normal subgroup of N. The group W = N/T is generated by a set S of elements of order 2 such that If s is an element of S and w is an element of W then sBw is contained in the union of BswB and BwB. No element of S normalizes B. The set S is uniquely determined by B and N and the pair (W,S) is a Coxeter system. Terminology BN pairs are closely related to reductive groups and the terminology in both subjects overlaps. The size of S is called the rank. We call B the (standard) Borel subgroup, T the (standard) Cartan subgroup, and W the Weyl group. A subgroup of G is called parabolic if it contains a conjugate of B, standard parabolic if, in fact, it contains B itself, and a Borel (or minimal parabolic) if it is a conjugate of B. Examples Abstract examples of BN pairs arise from certain group actions. Suppose that G is any doubly transitive permutation group on a set E with more than 2 elements. We let B be the subgroup of G fixing a point x, and we let N be the subgroup fixing or exchanging 2 points x and y. The subgroup T is then the set of elements fixing both x and y, and W has order 2 and its nontrivial element is represented by anything exchanging x and y. Conversely, if G has a (B, N) pair of rank 1, then the action of G on the cosets of B is doubly transitive. So BN pairs of rank 1 are more or less the same as doubly transitive actions on sets with more than 2 elements. More concrete examples of BN pairs can be found in reductive groups.