Borel subgroup is conjugate-dense in connected algebraic group

From Groupprops
Jump to: navigation, search

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

Statement over an algebraically closed field

Suppose G is a connected algebraic group over an algebraically closed field K and B is a Borel subgroup of G. Then, B is a conjugate-dense subgroup of G, i.e., we can write:

G = \bigcup_{g \in G} gBg^{-1}

Statement over an arbitrary field

The statement as made over an algebraically closed field does not work over an arbitrary field, but we can make suitable modifications and make sure it does (can we?).

Particular cases

Theorem Value of G Value of B
Triangulability theorem general linear group subgroup of upper-triangular matrices.
Spectral theorem for unitary matrices (not sure if this really fits) unitary group subgroup of unitary diagonal matrices.