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.