If $G$ acts primitive on $\Omega$ and $\alpha, \beta \in \Omega$ are two distinct points, then $G_{\alpha} \ne G_{\beta}$.
It might be simple, but I do not see why it holds? I guess if $G_{\alpha} = G_{\beta}$, then $\Delta := \{\alpha,\beta\}$ is a candidate for block, of course we have $\alpha^x = \alpha \Leftrightarrow \beta^x = \beta$, but if $\alpha^x = \beta$, then I cannot establish $\beta^x = \alpha$ (I guess this must hold, at least it implies that $\Delta$ would be a block)?