Stabilizers of Elements in Same Orbit are Conjugate Subgroups
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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.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54 \beta$