# Borel subgroup of general linear group

*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*

## Contents

## Definition

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

- It is the subgroup of (the general linear group: the group of invertible matrices over ) comprising the upper-triangular matrices.
- It is the subgroup of 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 is an algebraically closed field, is *conjugate-dense* in : every element of is conjugate to some element in . 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 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`