# 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 \ \$ $\ds z$ $\in$ $\ds \set {g \in G: g \circ x = x \circ g}$

$\blacksquare$