Stabilizer of Element under Conjugacy Action is Centralizer
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $*$ be the conjugacy action on $G$ defined by the rule:
- $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$
Let $x \in G$.
Then the stabilizer of $x$ under this conjugacy action is:
- $\Stab x = \map {C_G} x$
where $\map {C_G} x$ is the centralizer of $x$ in $G$.
Proof
From the definition of centralizer:
- $\map {C_G} x = \set {g \in G: g \circ x = x \circ g}$
Then:
\(\ds z\) | \(\in\) | \(\ds \Stab x\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds z\) | \(\in\) | \(\ds \set {g \in G: g \circ x \circ g^{-1} = x}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds z\) | \(\in\) | \(\ds \set {g \in G: g \circ x = x \circ g}\) |
$\blacksquare$
Also see
- Conjugacy Action on Group Elements is Group Action
- Orbit of Element under Conjugacy Action is Conjugacy Class
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $108$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$