How $\bigcap_{s\in S} G_s$ came ? . Why not $\bigcup_{s\in S} G_s $? Relation between stablizers and  kernels of  group actions
Stablizers :let $s$ is some  fixed element of $S $ then  the stabilizier of $s$ in $G$ is defined by $G_s=\{g \in G | g.s=s\}$
Kernel of action of  $G$ on $S$ is defined as
$$\left\{\,g\in G\;:\;\forall\,s\in S\;,\;\;gs=s\,\right\}=\left\{\;g\in G\;;\;g\in G_s\;,\;\;\forall\,s\in S\;\right\}=\bigcap_{s\in S} G_s$$
My  confusion: Im not  getting how $\bigcap_{s\in S} G_s$ came . Why  not $\bigcup_{s\in S} G_s $?
 A: First, one should never write "$\forall s\in S$" after the condition to which it applies: formulas have there rules, to which one should obey, independently of the rules of natural language that apply outside formulas (so the fact that "for all $s$" could be added in English as an afterthought does not legitimize writing it to the right in a formula, no more that the fact that one could say "the values of $f$ at $s^2$, summed over all $s$"  in English justifies writing $f(s^2)\sum_{s\in S}$).
Then to the meat of the question, $\bigcap$ generalises the $\cap$ operator for intersection, which is related to the logical "and" operator $\land$ (since $A\cap B$ is $\{\,x\in U\mid x\in A\land x\in B\,\}$ when $U$ contains both $A$ and $B$), and the generalisation of $\land$ to collections of conditions parametrised over $s$ in a set $S$ is $\forall s\in S$. Everything is natural here, except for the pointy part of $\forall$ pointing down instead of up, which is an unfortunate consequence of historic accidents. Please don't infer from that direction that "for all" goes with the operations $\bigcup$, $\cup$ (union) and $\lor$ ("or"): the operator that does that is $\exists s\in S$ instead. In summary, the $\forall s\in S$ should have made you think of $\bigcap_{s\in S}$, and certainly not of $\bigcup_{s\in S}$; indeed $\bigcap_{s\in S}G_s$ is precisely, and by definition, $\{\, g\in G\mid \forall s\in S:g\in G_s\,\}$.
