(B,N) pair

In mathematics, a (B, N) pair is a structure on groups of Lie type

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that allows one to give uniform proofs of many results, instead of giving a large number of

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case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the

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general linear group over a field. They were invented by the mathematician Jacques Tits, and are also sometimes known as Tits systems.

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A (B, N) pair is a pair of subgroups B and N of a group G such that the following axioms hold:

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• 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 of elements w_i of order 2, for i in some non-empty set I.
• If w_i is one of the generators of W and w is any element of W, then w_iBw is contained in the union of Bw_iwB and BwB.
• No generator w_i normalizes B.

The idea of this definition is that B is an analogue of the upper triangular matrices of

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the general linear group GLn(K), H is an analogue of the diagonal matrices,

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and N is an analogue of the normalizer of H.

(B,N) pair: Definition ►



 

Groups of Lie type: REDIRECT Group of Lie type...

General linear group: In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. (This is indeed a group because the product of two invertible...

Jacques Tits: Jacques Tits (born August 12, 1930) is a Belgian mathematician. He has written and cowritten a large number of papers on a number of subjects, principally algebra....




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