Triangulability theorem: Difference between revisions

Latest revision as of 18:49, 7 September 2008

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, Conjugate-dense subgroup (?)) in a particular group or type of group (namely, General linear group (?)).

This article gives the statement, and proof, of a particular subgroup in a group being conjugate-dense: in other words, every element of the group is conjugate to some element of the subgroup

Statement

Let $k$ be an algebraically closed field. Denote by $G L_{n} (k)$ the general linear group of invertible $n \times n$ matrices over $k$ and by $B_{n} (k)$ the Borel subgroup: the subgroup of invertible upper triangular matrices. Then, $B (n, k)$ is conjugate-dense in $G L (n, k)$ . In other words, given any matrix in $G L (n, k)$ , we can conjugate it (or change basis) to make it upper triangular.

This is an immediate corollary of the Jordan canonical form theorem, though it can be proved independently as well.

Note that the result fails over non-algebraically closed fields; in particular, it fails when $k$ is a finite field.

@@ Line 1: / Line 1: @@
+{{subgroup property satisfaction|
+subgroup = Borel subgroup in general linear group|
+group = general linear group|
+property = conjugate-dense subgroup}}
 {{conjugate-dense subgroup statement}}
 ==Statement==
+Let <math>k</math> be an [[algebraically closed field]]. Denote by <math>GL_n(k)</math> the [[general linear group]] of invertible <math>n \times n</math> matrices over <math>k</math> and by <math>B_n(k)</math> the [[fact about::Borel subgroup in general linear group|Borel subgroup]]: the subgroup of invertible upper triangular matrices. Then, <math>B(n,k)</math> is conjugate-dense in <math>GL(n,k)</math>. In other words, given any matrix in <math>GL(n,k)</math>, we can conjugate it (or change basis) to make it upper triangular.
+This is an immediate corollary of the [[Jordan canonical form theorem]], though it can be proved independently as well.
+Note that the result fails over non-algebraically closed fields; in particular, it fails when <math>k</math> is a [[finite field]].