Let $p: G \rightarrow Sym(X)$ denote the permutation representation of the G-set X. Show that $ker(p)=\bigcap_{x \in X} Stab_G(x)$
Approach: I am finding the notation of this proof a little bit challenging if done by using element wise proof. I am just wondering if we just can give a combinatorial argument, but the combinatorial argument is also looking hard. I am having difficulties understanding how $p: G \rightarrow Sym(X)$ behaves. I feel like it can't be viewed as a pure function since there is another function going on at the end. If we let $y \in ker(p)$, then that would imply $p(y)=I_x$, but then the identity takes an element $g \in G$ and map it in the form $g*x=x$