On primitive permutation groups
Let $G$ be a primitive permutation group on a finite set $X$, $x\in X$, $y\in X \ \{y\}$ and $G_{x,y}\trianglerighted G_x$. It is proved that, if $G$ is of type I, type III(a), type III(c) (of the O'Nan-Scott classification) or $G$ is of type II and $soc(G)$ is not an exceptional group of Lie type or a sporadic simple group, then $G_{x,y}=1$. In addition, it is proved that if G is of type III(b) and $soc(G)$ is not a direct product of exceptional groups of Lie type or sporadic simple groups, then $G_{x,y}=1$.