Show that the intersections of the $G_s$ is normal subgroup of $G$ I need to prove that given a  group $G$ acting in a set $S$, the intersection of the stabilizers $G_s$, where $G_s:=\{g\in G: g.s=s\}$ and $s$ varies through all $S$, is a normal subgroup of $G$.
But the thing is to find the right homomorphism, or is there an easier way to do it?, Can you help me to prove this result please? Thank you.    

Let $G_a$ be the stabilizer of $a$. It is easy to see $gG_ag^{-1}$ is a subgroup of the stabilizer of $ga$ (In fact they are equal but you don't need this).
So any conjugate of a stabilizer is a subgroup of another stabilizer, thus the image of the intersection of all stabilizers under conjugation by $g$  is a subgroup of the intersection of all stabilizers, in other words the intersection of the stabilizers is normal.
