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 subgroupsB and N of a group G such that the following axioms hold:
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 (B, N) 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 (B, N) 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 (B, N) pairs can be found in reductive groups.
Suppose that G is the general linear group GLnK over a field K. We take B to be the upper triangularmatrices, 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 (B, N) 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.[2]
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. 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.[5] More generally, this bijection extends to conjugacy classes of parabolic subgroups.[6]
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.
