\begin{definition}[Definition:Conjugacy Action/Subsets]
Let $G$ be a group.
Let $\powerset G$ be the power set of $G$.
The '''(left) conjugacy action on subsets''' is the group action $* : G \times \powerset G \to \powerset G$:
:$g * S = g \circ S \circ g^{-1}$
The '''right conjugacy action on subsets''' is the group action $* : \powerset G \times G \to \powerset G$:
:$S * g = g^{-1} \circ S \circ g$
\end{definition}