\begin{definition}[Definition:Conjugacy Action/Subgroups]
Let $X$ be the set of all subgroups of $G$.
The '''(left) conjugacy action on subgroups''' is the group action $* : G \times X \to X$:
:$g * H = g \circ H \circ g^{-1}$
The '''right conjugacy action on subgroups''' is the group action $* : X \times G \to X$:
:$H * g = g^{-1} \circ H \circ g$
\end{definition}