I am reading Vinberg's algebra text, and on page 144 he says "Of course, not every transformation group leads to a geometry which is interesting and also important for some applications. All such geometries are connected to quite rich transformation groups, and there are not many of them. The minimal condition here is transitivity."
He defines a group G to be a transitive group of some set X if for any two points in X there exists an element of G taking one to the other. So in that sense, all points are congruent.
I am wondering about how transitive groups give rise to a "geometry". For example, SO(3) would be a transitive group of the sphere, but not a transitive group of three-space. Can someone explain this concept perhaps using SO(3) as an example and a non-example via those two sets, the sphere and three-space?