https://www.encyclopediaofmath.org/index.php/O'Nan-Scott_theorem
Here it says that the stabliser is a maximal subgroup of $G$ (which I understand why) containing no non-trivial normal subgroup of $G$, why is that?
Suppose $G_a$ contains $N$ where $N\lhd G$, then $G=HN$ for some subgroup $H$, then $H$ is transitive and hence primitive. What else can I say? what leads to a contradiction so we can't have $N$ non-trivial?
One extra question is that how is this statement connected to minimal normal subgroups in the proof of the O'Nan-Scott theorem?