Borel subgroup is abnormal in general linear group

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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 (?)).


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).

Related facts

For finite fields

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 1s 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