While a group is fully defined by the structure of the group manifold $\,M\,$, in applications a group is usually introduced as a set of transformations acting on some non-empty set $\,\mathbb{H}\;$ — and the structure of this set is of a practical interest.
Definition 1. $\;\;$ For an arbitrary point $\,{{p}}\in\mathbb{H}\,$, the set of all points of $\,\mathbb{H}\,$, to which $\,{{p}}\,$ can be mapped by the elements of $\,G\,$, is called the orbit of $\,{{p}}\,$ and is denoted as $\,G\,{{p}}\;$:
$$
G{{p}}\;=\;\left\{g\,p\right\}\;\,.
$$
Definition 2. $\;\;$ A group action $\,G\,\times\,\mathbb{H}\,\longrightarrow\,\mathbb{H}\,$ is transitive if it spans a single group orbit, i.e., if for every pair of points $\,p,\,q\,$ on that orbit there is a group element $\,g\,\in\,G\,$ such that $\,g\,p=q\,$.
Definition 3. $\;\;$ A set with a transitive action by a group is called homogeneous space.
Now, question:
From the above definitions, I understand that an orbit is a homogeneous space. Is a homogeneous space always an orbit? Are these two notions synonyms? Or is there a difference between them?
Best Answer
A homogenous space $\mathbb H$ is a topological space for which, for every $x,y\in\mathbb H$ there is a homeomorphism $\phi:\mathbb H\to\mathbb H$ so that $\phi(x)=y.$ This is more a topology definition, rather than a group definition.
However, given any topological space, $\mathbb H$, the set of homeomorphisms $\mathbb H\to\mathbb H$ form a group, and that group acts transitively if and only if $\mathbb H$ is homogeneous, so every homogeneous space has a group that acts on $\mathbb H$ transitively, and thus $\mathbb H$ is an orbit.
Being homogeneous means, roughly, that from a topological point of view, any point in the space $\mathbb H$ looks like any other point in the space. So $\mathbb R^k$ and the circle are homogeneous, but the space of two lines intersecting at a point is not homogeneous, because the intersection point is not "like" the other points in that space.