(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.
A ("B", "N") pair is a pair of subgroups "B" and "N" of a group "G" such that the following axioms hold:
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.
Properties of groups with a BN pair.
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.
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.