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From Wikipedia, the free encyclopedia
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.
A (B, N) pair is a pair of subgroups B and N of a group G such that the following axioms hold:
The set S is uniquely determined by B and N and the pair (W,S) is a Coxeter system.[1]
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
A subgroup of G is called
Abstract examples of (B, N) pairs arise from certain group actions.
More concrete examples of (B, N) pairs can be found in reductive groups.
The Bruhat decomposition states that G = BWB. More precisely, the double cosets B\G/B are represented by a set of lifts of W to N.[3]
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]
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.
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