Definition:Conjugacy Action

Definition
Let $\left({G, \circ}\right)$ be a group.

The (left) conjugacy action of $G$ is the left group action $* : G\times G \to G$ defined as:
 * $\forall g, x \in G: g * x = g \circ x \circ g^{-1}$

The right conjugacy action of $G$ is the right group action $* : G\times G \to G$ defined as:
 * $\forall x, g \in G: x * g = g^{-1} \circ x \circ g$

Also see

 * Conjugacy Action is Group Action