For instance, translations form a commutative subgroup that acts freely and transitively on, while the stabilizer of any point there is the aforementioned.
In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices.
The stabilizer subgroup of a flag acts simply transitively on adapted bases for the flag, and thus these are not unique unless the stabilizer is trivial.
