I have a question, possibly silly, possibly already asked.
Imagine I have a Lie group $G$ such that $G$ quotient with its center, $Z(G)$ is a Lie group. (Bonus question: Is this not always the case and if so, is there a nice counterexample?).
If $G$ acts transitively on a manifold $M$, is it true $G/Z(G)$ inherits the action on $M$ and that it is also transitive?
If $G$ acts transitively on a sub-manifold (not subgroup!) $N \subset G$ by some action, does $G/Z(G)$ act transitively on $N/Z(G)$ by the "natural" inherited action?