Pair ( B , N )

A pair ( B , N ) is a pair of subgroups B and N of a group G satisfying axioms ^[1]

• The union of the groups B and N generates G.
• The intersection H of the groups B and N is a normal subgroup of the group N.
• The group W = N / H is generated by the set S of elements w _{i of} order 2 for i in some nonempty set I.
• If w _i is an element of S and w is any element of the group W , then w _i Bw is contained in the union of Bw _i wB and BwB .
• No generator w _i normalizes B.

The idea of ​​the definition is that B is an analog of the upper triangular matrices of the complete linear group GL _n ( K ), H is an analog of diagonal matrices, and N is an analog of the normalizer H.

The subgroup B is sometimes called the , 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 rank .

• Suppose that G is any on a set X with more than two elements. Let B be a subgroup of G leaving the point x in place, and let N be a subgroup leaving in place or swapping 2 points x and y . The subgroup H then consists of elements leaving both points x and y in place, and W has order 2 and its nontrivial element permutes x and y .
• Conversely, if G has a pair (B, N) of rank 1, then the action of G on the adjacency classes of B is . Thus, BN pairs of rank 1, more or less, are the same as the action of double permutation actions on a set of more than 2 elements.
• Suppose that G is a complete linear group GL _n ( K ) over a field K. We take upper triangular matrices as B , diagonal matrices as H , and N] as N , that is, matrices with exactly one nonzero entry in each column and in each row. There are n - 1 generators w _i represented by matrices obtained by rearranging the rows of the diagonal matrix.
• More generally, any group of Lie type has a BN pair.
• A redgebraic group over a local field has a BN-pair, where B is an Iwahori subgroup .

The mapping w into BwB is an isomorphism from the set of elements of W to the set of double cosets of the group G along B. Classes form the G = BWB .

If T is a subset of the group S , let W ( T ) be the subgroup of W generated by the subset T. We define G ( T ) = BW ( T ) B as the standard for T. Subgroups of G containing conjugate subgroups of B are parabolic subgroups ^[2] . Adjacent classes of B are called (or minimal parabolic subgroups). These are exactly the standard parabolic subgroups.

BN pairs can be used to prove that many groups of Lie type are simple modulo centers. More precisely, if G has a BN- pair such that B is solvable , the intersection of all adjacent classes of B is trivial, and the set of generators of W cannot be decomposed into two non-empty commuting sets, then G is simple if it is (then is the same as your commutant ). In practice, all these conditions, with the exception of the perfection of the group G , are easy to verify. Checking the perfection of the group G requires some complicated calculations (and some small groups of Lie type are not perfect). However, to show that the group is perfect is usually much easier than to show that the group is simple.