Borel subgroup of general linear group

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This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

Let K be a field and n be a natural number. The Borel subgroup B(n,K), also called the Borel subgroup in general linear group, is defined in the following equivalent ways:

  • It is the subgroup of GL(n,K) (the general linear group: the group of invertible n \times n matrices over K) comprising the upper-triangular matrices.
  • It is the subgroup of GL(n,K) comprising those linear transformations that preserve the standard complete flag of subspaces.

Properties

Conjugate-dense subgroup for algebraically closed fields

Further information: Triangulability theorem

When K is an algebraically closed field, B(n,K) is conjugate-dense in GL(n,K): every element of GL(n,K) is conjugate to some element in B(n,K). This is a weak version of the Jordan canonical form theorem.

Self-normalizing subgroup

Further information: Borel subgroup is self-normalizing in general linear group

Bruhat decomposition

Further information: Bruhat decomposition theorem

The double coset space of the Borel subgroup is parametrized by the symmetric group on n letters; their permutation matrices act as double coset representatives.

Abnormal subgroup

Further information: Borel subgroup is abnormal in general linear group

Borel subgroup in the algebraic sense

Further information: Borel subgroup in general linear group is a Borel subgroup in the algebraic sense

In other words, it is a maximal connected solvable algebraic subgroup.

Normalizer of upper-triangular unipotent subgroup

Further information: Borel subgroup equals normalizer of upper-triangular unipotent subgroup

Note that, for a finite field, the upper-triangular unipotent subgroup is a Sylow subgroup, so the Borel subgroup is the normalizer of a Sylow subgroup. this gives an alternate explanation for its being an abnormal, and in particular, a self-normalizing subgroup.

Description of subgroups containing it

Further information: Parabolic subgroups of the general linear group