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.