\begin{definition}[Definition:Conjugacy Action]
Let $\struct {G, \circ}$ 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:
:$\forall x, g \in G: x * g = g^{-1} \circ x \circ g$
\end{definition}