Conjugacy Action is Group Action

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Action on Group Elements
$G$ acts on itself by the rule $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$.

Also:
 * $\operatorname{Stab} \left({x}\right) = C_G \left({x}\right)$, where $C_G \left({x}\right)$ is the centralizer of $x$ in $G$.


 * $\operatorname{Orb} \left({x}\right) = C_{x}$, where $C_{x}$ is the conjugacy class of $x$.

Action on Subgroups
Let $X$ be the set of all subgroups of $G$.

For any $H \le G$ and for any $g \in G$, we define: $\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$

This is a group action.

Also:
 * $\operatorname{Stab} \left({H}\right) = N_G \left({H}\right)$ where $N_G \left({H}\right)$ is the normalizer of $H$ in $G$.


 * $\operatorname{Orb} \left({H}\right)$ is the set of subgroups conjugate to $H$.

Elements

 * Clearly GA-1 is fulfilled as $e * x = x$.


 * GA-2 is shown to be fulfilled thus:


 * $\operatorname{Stab} \left({x}\right) = C_G \left({x}\right)$ follows from the definition of centralizer: $C_G \left({x}\right) = \left\{{g \in G: g \circ x = x \circ g}\right\}$.

Furthermore, since the powers of $x$ commute with $x$, $\left \langle {x} \right \rangle \in C_G \left({x}\right)$.


 * $\operatorname{Orb} \left({x}\right) = C_{x}$ follows from the definition of the conjugacy class.

Subgroups

 * Clearly GA-1 is fulfilled as $e * H = H$.


 * GA-2 is shown to be fulfilled thus:


 * $\operatorname{Stab} \left({H}\right) = \left\{{g \in G: g \circ H \circ g^{-1} = H}\right\}$ which is how the normalizer is defined.


 * $\operatorname{Orb} \left({H}\right) = \left\{{g \circ H \circ g^{-1}: g \in G}\right\}$ from the definition.