Borel subgroup of general linear group

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.

Conjugate-dense subgroup for algebraically closed fields
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.

Bruhat decomposition
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.

Borel subgroup in the algebraic sense
In other words, it is a maximal connected solvable algebraic subgroup.

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.