Kernel of Group Action is Normal Subgroup

Theorem
Let $$G$$ be a group of transformations of a set $$X$$.

The set $$G_0$$ defined as $$G_0 = \left\{{g \in G: \forall x \in X: g \wedge x = x}\right\}$$ is a normal subgroup of $$G$$.

Proof
Let $$h \in G_0$$.

$$ $$ $$ $$ $$ $$

Thus $$G_0$$ is the intersection of subgroups ($$\operatorname{Stab} \left({x}\right) \le G$$ from Stabilizer is Subgroup) and by Intersection of Subgroups: Generalized Result, $$G_0 \le G$$.


 * Now we need to prove normality.