Algebraic Group Actions, Orbits, and Stabilizers

Let \(G\) be a linear algebraic group and \(X\) an algebraic variety. A regular action of \(G\) on \(X\) is a morphism \[ G \times X \to X,\qquad (g,x)\mapsto g x, \] such that \(e x=x\) and \(g(hx)=(gh)x\) for all points.

For \(x \in X\), the orbit and stabilizer are \[ Gx=\{gx:g\in G\},\qquad G_x=\{g\in G:gx=x\}. \] The stabilizer \(G_x\) is a closed algebraic subgroup of \(G\), since it is the fiber over \(x\) of the orbit morphism \(G\to X\), \(g\mapsto gx\).

The action is transitive if \(X=Gx\) for some, equivalently every, \(x\in X\). In that case \(X\) is a homogeneous space for \(G\), and the stabilizer records the quotient description \(X \simeq G/G_x\) whenever the geometric quotient exists.