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

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