Homogeneous space

Remarks

Let $x \in X(k)$ and let $G_x$ be its stabilizer, a closed subgroup of $G$. The orbit map $g \mapsto g \cdot x$ induces a bijective morphism $G / G_x \to X$. In characteristic $0$ this morphism is an isomorphism, so $X \cong G / G_x$. In characteristic $p$ it can fail to be an isomorphism, because the orbit map need not be separable.

Example: $\mathrm{GL}_{n+1}$ acts on $\mathbb{P}^n$ by $g \cdot [v] = [g v]$. Any nonzero vector can be sent to any other by an invertible matrix, so the action is transitive and $\mathbb{P}^n$ is a homogeneous space.

Non-example: $\mathbb{G}_m$ acts on $\mathbb{A}^1$ by $t \cdot x = t x$. There are two orbits, $\set{0}$ and $\mathbb{A}^1 \setminus \set{0}$, so the action is not transitive.

In other settings, such as Lie groups and topological groups, a homogeneous space is a manifold or a space with a transitive action of the group.