Least dimension of Lie group acting transitively of a manifold

Question

I guess that the least possible dimension of a Lie group $G$ acting smoothly and transitively on a compact manifold $M$ is $\operatorname{dim}(M)$.

Is this correct and is there a ref?

Answer 1

This follows from Sard's theorem. If $G \times M \to M$ is smooth, then in particular $G \times \{pt\} \to M$ is smooth. If $\operatorname{dim} G < \operatorname{dim} M$, then Sard's theorem implies the image has measure zero, and in particular is not all of $M$.

Compactness is inessential here.