Stabilizers are a subgroup

up:: Stabilizer of Group Action

Let there be a Group Action of $G$ over $X$, and let $x\in X$. Then the stabilizer of $x$ is

$$Sta{b}_x=\{g\in G\mid g\cdot x=x\}\subseteq G$$

1. Operation closure

Let $g,h\in Sta{b}_x$. Then

$$gh\cdot x=g\cdot (h\cdot x)=g\cdot x=x$$

which implies that $gh\in Sta{b}_x$

2. Identity element

Let $e_G\in Sta{b}_x$, since $e_G\cdot x=x$.

3. Inverse element

Let $g\in Sta{b}_x$. Then, since

$$x=e_G\cdot x=g^{-1}\cdot g\cdot x=g^{-1}\cdot x$$

Thus, $g\in Sta{b}_x⟹g^{-1}\in Sta{b}_x$.

Thus, $Sta{b}_x\le G$ is a Normal Subgroup of $G$.