# 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:

 $\ds \Stab y$ $=$ $\ds \set {g \in G: g * y = y}$ Definition of Stabilizer $\ds$ $=$ $\ds \set {g \in G: g * \paren {h_1 h_2^{-1} * z} = h_1 h_2^{-1} * z}$ $\ds$ $=$ $\ds \set {g \in G: h_1^{-1} h_2 * \paren {g h_1 h_2^{-1} * z} = z}$ $\ds$ $=$ $\ds \set {g \in G: \paren {h_1 h_2^{-1} }^{-1} g \paren {h_1 h_2^{-1} } * z = z}$ $\ds$ $=$ $\ds \paren {h_1 h_2^{-1} } \set {\paren {h_1 h_2^{-1} }^{-1} g \paren {h_1 h_2^{-1} } \in G: \paren {h_1 h_2^{-1} }^{-1} g \paren {h_1 h_2^{-1} } * z = z} \paren {h_1 h_2^{-1} }^{-1}$ $\ds$ $=$ $\ds \paren {h_1 h_2^{-1} } \Stab z \paren {h_1 h_2^{-1} }^{-1}$ Definition of Stabilizer

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

Hence the result.

$\blacksquare$