# (*B*, *N*) pair

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[edit]

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,
*H*, of*B*and*N*is a normal subgroup of*N*. - The group
*W*=*N/H*is generated by a set*S*of elements*w*of order 2, for_{i}*i*in some non-empty set*I*. - If
*w*is an element of_{i}*S*and*w*is any element of*W*, then*w*is contained in the union of_{i}Bw*Bw*and_{i}wB*BwB*. - No generator
*w*normalizes_{i}*B*.

The idea of this definition is that *B* is an analogue of the upper triangular matrices of
the general linear group *GL*_{n}(*K*), *H* is an analogue of the diagonal matrices, and *N* is an analogue of the normalizer of *H*.

The subgroup *B* is sometimes called the **Borel subgroup**, *H* is sometimes called the **Cartan subgroup**, and *W* is called the **Weyl group**. The pair (*W*,*S*) is a *Coxeter system*.

The number of generators is called the **rank**.

## Examples[edit]

- Suppose that
*G*is any doubly transitive permutation group on a set*X*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*H*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. - Suppose that
*G*is the general linear group*GL*_{n}(*K*) over a field*K*. We take*B*to be the upper triangular matrices,*H*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*w*, represented by the matrices obtained by swapping two adjacent rows of a diagonal matrix._{i} - More generally, any group of Lie type has the structure of a BN-pair.
- A reductive algebraic group over a local field has a BN-pair where
*B*is an Iwahori subgroup.

## Properties of groups with a BN pair[edit]

The map taking *w* to *BwB* is an isomorphism from the set of elements of *W* to the set of double cosets of *B*; this is the Bruhat decomposition *G* = *BWB*.

If *T* is a subset of *S* then let *W*(*T*) be the subgroup of *W* generated by *T*: we define and *G*(*T*) = *BW*(*T*)*B* to be the *standard parabolic subgroup* for *T*.
The subgroups of *G* containing conjugates of *B* are the *parabolic subgroups*; conjugates of *B* are called Borel subgroups (or minimal parabolic subgroups). These are precisely the standard parabolic subgroups.

## Applications[edit]

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.

## References[edit]

- Bourbaki, Nicolas (2002).
*Lie Groups and Lie Algebras: Chapters 4–6*. Elements of Mathematics. Springer. ISBN 3-540-42650-7. Zbl 0983.17001. The standard reference for BN pairs. - Serre, Jean-Pierre (2003).
*Trees*. Springer. ISBN 3-540-44237-5. Zbl 1013.20001.