Conjugacy Action on Group Elements is Group Action

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

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

is a group action on itself.

Proof
We have that:
 * $e * x = e \circ x \circ e^{-1} = x$

and so is fulfilled.

is shown to be fulfilled thus:

Also see

 * Stabilizer of Element under Conjugacy Action is Centralizer
 * Orbit of Element under Conjugacy Action is Conjugacy Class