I am new here and don't know much about Latex so, I attach my question from Permutation Groups by J. Dixon. I hope to get a help for it:
1.6.5 Let $G$ be a transitive subgroup of $\mathrm{Sym}(\Omega)$ and let $\alpha \in \Omega$. Show that $\mathrm{fix}(G_\alpha)$ is a block for $G$. In particular, if $G$ is primitive, then either $\mathrm{fix}(G_\alpha)=\{\alpha\}$ or else $G_\alpha=1$ and $G$ has prime degree.
Thank you.