Let G = GL_2(F) for some field F and B be the subgroup of upper triangular matrices in G. As usual, let \mathbb P^1 = \mathbb P^1(F) denote the projective plane over F.

As F-varieties, G/B is isomorphic to \mathbb P^1 because B is the stabilizer of the usual (Mobius) action of G on \mathbb P^1. The resulting action of G/B on \mathbb P^1 is 1-transitive and fixed-point free. Indeed, counting points when F is a finite field with q elements agrees with the facts that

\# G(\mathbb F_q) = (q^2-1)(q^2-q)

and

\# \mathbb P^1(\mathbb F_q) = q+1.

This generalizes in that parabolic subgroups (closed subgroup containing a Borel subgroup, i.e., containing the upper triangular matrices upto conjugation, for the general linear group) of an algebraic group give rise to projective varieties. They are precisely the stabilizers of a flag and conversely, the set of flags can be given the structure of an F-space and in particular a topology wherein they are compact spaces (being projective spaces). They are called flag varieties.

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