# Borel subgroup is abnormal in general linear group

Let $k$ be a field, and $GL_n(k)$ denote the General linear group (?): the group of invertible $n \times n$ matrices over $k$. Let $B_n(k)$ denote the Borel subgroup of $Gl_n(k)$: the subgroup of invertible upper-triangular matrices. Then, $B_n(k)$ is an Abnormal subgroup (?) inside $GL_n(k)$.
Suppose $k$ is a finite field of order $q$, where $q$ is the power of a prime $p$. Then, the Borel subgroup $B_n(k)$ is the normalizer of a $p$-Sylow subgroup of $GL_n(k)$: the subgroup of upper-triangular matrices with $1$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