Stabilizer of Element under Conjugacy Action is Centralizer

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 $10.10$