Group action on a manifold with finitely many orbits I'm looking for a result along the lines of the following: > Let $G$ be a group acting on a set $X$. If the action partitions $X$ into finitely many $G$-orbits, then $\dim G \geq \dim X$. For this to even make sense, it seems like $G$ and $X$ should have a vector space/manifold structure, but there may very well be additional assumptions for the conclusion to hold (for instance, $G$ is also an algebraic group, and the action is polynomial, etc.). Can someone provide a more complete statement/proof of this result? This result is used in Tits' argument in Gabriel's theorem (see for example page 29 of Bernstein, Gel'fand, and Ponomarev's article "Coxeter functors and Gabriel's theorem"), but I would also be interested to see how this result may be used in larger contexts.

Here is a proof in the smooth category. Let $\alpha : G \times X \to X$ be an action of a Lie group $G$ on a smooth manifold $X$. For $x \in X$, the image of the map $\alpha(-, x) : G \to X$ is the orbit containing $x$. If $\dim G \lt \dim X$, then by Sard's lemma the image of $\alpha(-, x)$ has measure zero for any $x$, and consequently $X$ cannot be the union of finitely or even countably many orbits.