Stabilizers of Elements in Same Orbit are Conjugate Subgroups

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


$y, z \in \Orb x$

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

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


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


\(\ds \Stab y\) \(=\) \(\ds \set {g \in G: g * y = y}\) 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.