Borel subgroup is self-normalizing in general linear group
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, Self-normalizing subgroup (?)) in a particular group or type of group (namely, General linear group (?)).
For finite fields
If is the finite field of order where is a power of a prime , the Borel subgroup equals the normalizer of a -Sylow subgroup of : the subgroup of upper-triangular matrices with s on the diagonal. The normalizer of any Sylow subgroup in a finite group is self-normalizing; in fact, it is an abnormal subgroup. Further information: Sylow normalizer implies abnormal
Hands-on proofPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Proof in terms of flags
The proof relies on a somewhat different interpretation of . Let denote the standard basis for . Then, the standard flag is an ascending chain of subspaces:
where the subspace is the span of the vectors . Then, is the subgroup of comprising those linear transformations that preserve this flag. In other words, if and only if the flag corresponding to the basis is the same as the flag corresponding to the basis .
Now, given any , the subgroup is precisely the subgroup of that stabilizes the flag of . Thus, for to normalize we require that the linear transformations stabilizing the flag of are the same as the linear transformations of the flag . We can now prove, through an inductive argument, that this forces the flags for the two bases to be the same, forcing .