Exercise of permutation groups

Question

Suppose that the group $G$ acts transitively on $\Omega$ and $\Gamma$ and $\Delta$ are finite subsets of $\Omega$ with $|\Gamma| < |\Delta| $. If $G_{(\Gamma)}$ and $G_{(\Delta)}$ act transitively on $\Omega \setminus \Gamma$ and $\Omega \setminus \Delta$, respectively, show that ${\Gamma}^x \subset \Delta$ for some $x \in G$. Does the result remain true if $\Gamma$ and $\Delta$ are infinite?

I think if $\Gamma \subset \Delta$ is trivial. I use induction on the members in intersection. But it doesn't work. I try to solve it but I stuck. I don't have any idea how to solve it. Any kind of suggestion is appreciated. Thanks to everyone for the help

Answer 1

You need to induct on $|\Gamma\setminus\Delta|$ rather than on $|\Gamma\cap\Delta|$.

Answer 2

For the infinite case, consider the group of all permutations of an uncountable set $\Omega$ with finite support (i.e. permutations that move only finitely many points).