Orbit of Element under Conjugacy Action is Conjugacy Class

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $*$ be the conjugacy group action on $G$:

$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

Let $x \in G$.

Then the orbit of $x$ under this group action is:

$\Orb x = C_x$

where $C_x$ is the conjugacy class of $x$.

Proof

Follows from the definition of the conjugacy class.

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$\blacksquare$

Also see

• Conjugacy Action on Group Elements is Group Action
• Stabilizer of Element under Conjugacy Action is Centralizer

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.15$