# Two basic definitions from Group Theory: homogeneous space and orbit.

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?

• It's not clear what you mean by group manifold. Are we talking about topological groups? If so, this might be worth mentioning since the words "basic definition" made me think the terms were beginning algebra definitions. Commented Mar 19, 2019 at 16:17
• Thomas Andrews, why is this detail important? My preference would be to stay within the realm of the simplest possible concepts. However, if the answer to my question depends on it, then we may as well assume that the group is topological. Commented Mar 19, 2019 at 16:24
• Observation: you need specify what is meant by "isomorphic" to make it meaningful. The questions also are not so clear, because of the setting being vague (I agree that the literature is often vague too about the meaning of "homogeneous space").
– YCor
Commented Mar 19, 2019 at 16:42
• It's not even clear what $G/(Gp)$ means, since transitivity means $Gp=\mathbb H,$ and $G/\mathbb H$ has no meaning. You might mean $G/G_p$, where $G_p=\{g\mid gp=p\}.$ Commented Mar 19, 2019 at 16:45
• Thomas, you are right. Thank you. In that Observation, I indeed was supposed to quotient out the group by the isotropy group of the point p. So I have now deleted that Observation as irrelevant. Commented Mar 19, 2019 at 16:55

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.
• I see from your answer that any topological space $\mathbb H$ is homogeneous with respect to its group of homeomorphisms, $\,\operatorname{Homeo}\,\mathbb H$, and therefore is an orbit under this group. My understanding is that under a subgroup $G < \,\operatorname{Homeo}\,\mathbb H$ the space $\mathbb H$ may split into separate orbits, correct? Then comes again my initial question, which so far remains unanswered: are homogeneous space and orbit synonyms? Thank you! Commented Sep 24, 2022 at 20:20