I am reading about permutation groups in "Finite Group Theory", Isaacs. Here the exercise 8A.5:
Let G act $k$-transitively on some set $\Omega$, and let $H=G_{\alpha}$, where $\alpha \in \Omega$. Let $\Delta$ be the sets of points in $\Omega$ that are fixed by $H$ (namely $Fix_H(\Omega)$). Show that $N_G(H)$ acts $r$-transitively on $\Delta$, where $r$ is the minimum of $k$ and $|\Delta|$.
If a group $G$ is $k$-transitive on $\Omega$ and $\alpha \in \Omega$, then the stabilizer $G_{\alpha}$ is $(k-1)$-transitive on $\Omega \setminus \{\alpha\}$, so in $\Omega \setminus \{\alpha\} $ there are no fixed points under $G_{\alpha}$, for $k \ge 2$. How the exercise should be? Should I take $k=1$?
There is something that I don't understand.