Stabilizer is Subgroup

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Theorem

Let $G$ be a group which acts on a set $X$.

Let $\operatorname{Stab} \left({x}\right)$ be the stabilizer of $x$ by $G$.

Then for each $x \in X$, $\operatorname{Stab} \left({x}\right)$ is a subgroup of $G$.

Let $G$ be a group whose identity is $e$, which acts on a set $X$.

Then:

$\forall x \in X: e \in \operatorname{Stab} \left({x}\right)$

Let $G$ be a group whose identity is $e$, which acts on a set $X$.

Then:

$\forall g, h \in G: g * x = h * x \iff g^{-1} h \in \operatorname{Stab} \left({x}\right)$

$\operatorname{Stab} \left({x}\right)$ can not be empty, because $e * x = x \implies e \in \operatorname{Stab} \left({x}\right)$.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle g, h$	$\in$	$\displaystyle \operatorname{Stab} \left({x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle g * x$	$=$	$\displaystyle x, h * x = x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle g * \left({h * x}\right)$	$=$	$\displaystyle x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \left({g h}\right) * x$	$=$	$\displaystyle x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle g h$	$\in$	$\displaystyle \operatorname{Stab} \left({x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Then $x = \left({g^{-1} g}\right) * x = g^{-1} * \left({g * x}\right) = g^{-1} * x$.

Hence $g^{-1} \in \operatorname{Stab} \left({x}\right)$.

Thus the conditions for the Two-step Subgroup Test are fulfilled, and $\operatorname{Stab} \left({x}\right) \le G$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle g, h\)	\(\in\)	\(\displaystyle \operatorname{Stab} \left({x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle g * x\)	\(=\)	\(\displaystyle x, h * x = x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle g * \left({h * x}\right)\)	\(=\)	\(\displaystyle x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \left({g h}\right) * x\)	\(=\)	\(\displaystyle x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle g h\)	\(\in\)	\(\displaystyle \operatorname{Stab} \left({x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)