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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 invented 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 ''GLn(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 number of generators ''wi'' is called the rank. EXAMPLES
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''. The subgroups of ''G'' containing ''B'' are called Parabolic Subgroup s. There are exactly 2n of them, and they correspond to subsets of ''I''. APPLICATIONS BN-pairs can be used to prove that most groups of Lie type are simple. 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 perfect. 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 or simple). But showing that a group is perfect is usually far easier than showing it is simple. FURTHER READING The standard reference for BN pairs is:
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