# Stabilizer is Subgroup/Corollary

## 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 \ \$ $\ds g^{-1} h$ $\in$ $\ds \Stab x$

$\blacksquare$