Orbit-Stabilizer Theorem/Proof 1

Proof
Let us define the mapping:
 * $\phi: G \to \Orb x$

such that:
 * $\map \phi g = g * x$

where $*$ denotes the group action.

It is clear that $\phi$ is surjective, because from the definition $x$ was acted on by all the elements of $G$.

Next, from Stabilizer is Subgroup: Corollary:
 * $\map \phi g = \map \phi h \iff g^{-1} h \in \Stab x$

This means:
 * $g \equiv h \pmod {\Stab x}$

Thus there is a well-defined bijection:
 * $G \mathbin / \Stab x \to \Orb x$

given by:
 * $g \, \Stab x \mapsto g * x$

So $\Orb x$ has the same number of elements as $G \mathbin / \Stab x$.

That is:
 * $\order {\Orb x} = \index G {\Stab x}$

The result follows.