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
- 1 Definition
- 2 Properties
- 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.
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.
Further information: Borel subgroup is self-normalizing in general linear group
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.
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