Thinking about how you can put vector fields on homogeneous spaces that respect the homogeneity, I'm interested in the following situation:
Let $V$ be a nonzero vector field on a manifold $M$, let the simply connected group $G$ act transitively on $M$, and let $V$ be invariant under this action, so $g_* V=V$ for all $g\in G$.
Now an obvious way to construct such a vector field is if $G$ acts freely and transitively, since you can pick a vector at any point and push it forward to any other point under the group action. If the action is not free, the ambiguity in choice of group element means that this may not work; the stabilizer need not map $V$ to itself. It is easy to come up with examples where this is not an obstruction, but in all those I've been able to think of, the action can be restricted to a free and transitive one. Must this always be the case? In other words, is the following true?
Proposition: Then $G$ has a subgroup acting freely and transitively.