Stabilizers of Elements in Same Orbit are Conjugate Subgroups

Theorem
Let $G$ be a group acting on a set $X$.

Let:
 * $y, z \in \Orb x$

where $\Orb x$ denotes the orbit of some $x \in X$.

Then their stabilizers $\Stab y$ and $\Stab z$ are conjugate subgroups.

Proof
From Stabilizer is Subgroup we have that both $\Stab y$ and $\Stab z$ are subgroups of $G$.

From definition of orbits:
 * $\exists h_1, h_2 \in G: y = h_1 * x, z = h_2 * x$

Then $y = h_1 * \paren {h_2^{-1} * z} = h_1 h_2^{-1} * z$.

Thus:

This shows that $\Stab y$ and $\Stab z$ are conjugate.

Hence the result.