Stabilizer is Subgroup/Corollary

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Corollary to Stabilizer is Subgroup

Let $G$ be a group whose identity is $e$.

Let $G$ act on a set $X$.

Let $x \in X$.


Then:

$\forall g, h \in G: g * x = h * x \iff g^{-1} h \in \Stab x$


Proof

\(\ds h * x\) \(=\) \(\ds g * x\)
\(\ds \leadstoandfrom \ \ \) \(\ds g^{-1} * \paren {h * x}\) \(=\) \(\ds g^{-1} * \paren {g * x}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {g^{-1} h} * x\) \(=\) \(\ds \paren {g^{-1} g} * x\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {g^{-1} h} * x\) \(=\) \(\ds e * x\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {g^{-1} h} * x\) \(=\) \(\ds x\)
\(\ds \leadstoandfrom \ \ \) \(\ds g^{-1} h\) \(\in\) \(\ds \Stab x\)

$\blacksquare$


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