Every orbit is an invariant subset of X on which G acts transitively.

This group acts transitively on all vertices, edges and triangles in the building.

That is to say, the general linear group acts transitively on the set of all complete flags.

The spin group acts transitively on S by rotations.

The important fact to note above is that the unitary group acts transitively on pure states.

The topological description is complicated by the fact that the unitary group does not act transitively on density operators.

Let G act transitively on X and pick some point .

A homogeneous space is a G-space on which G acts transitively.

It is simple linear algebra to show that GL acts transitively on those.

Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices.