# Borel subgroup is abnormal in general linear group

From Groupprops

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Borel subgroup in general linear group (?)) satisfying a particular subgroup property (namely, Abnormal subgroup (?)) in a particular group or type of group (namely, General linear group (?)).

## Statement

Let be a field, and denote the General linear group (?): the group of invertible matrices over . Let denote the Borel subgroup of : the subgroup of invertible upper-triangular matrices. Then, is an Abnormal subgroup (?) inside .

## Related facts

### For finite fields

Suppose is a finite field of order , where is the power of a prime . Then, the Borel subgroup is the normalizer of a -Sylow subgroup of : the subgroup of upper-triangular matrices with s on the diagonal. The abnormality in this case follows from the general fact that the normalizer of a Sylow subgroup is abnormal. `Further information: Sylow normalizer implies abnormal`