Let $H$ be a subgroup of a finite group $G$. Let $A$, $B$ $\in \mathcal{P}(G)$. Define $A$ to be conjugate to $B$ with respect to $H$ if $B=hAh^{-1}$ for some $h\in H$. Then find orbit and stabilizer with respect to conjugation.
Solution :
We take $X=\mathcal{P}(G)$ and H to be the actor. We say conjugation is an action on $\mathcal{P}(G)$. Also it is equivalence relation (basic information). For the orbit we write:
$$C_{H}(A)=\lbrace \phi(h*A):h\in H \rbrace,\; \text{where } \phi \text{ is action on } \mathcal{P}(G)= \\ C_{H}(A)=\lbrace hAh^{-1} :h\in H \rbrace$$
For the stabilizer we have:
$$H_{A}=\lbrace h\in H:hAh^{-1}=A\rbrace$$
And then how can I move on? Textbook says $H_{A}=H \cap N(A)$ where $N(A)$ is a normalizer of $A$
Thanks in advance!
The group action here $H×X\to X $ defined by $$h\cdot A=hAh^{-1}$$
Orbit:
$\mathcal{O}_A=\{h\cdot A :h\in H\}=C_H(A)$
Stabilizer:
$\begin{align}\operatorname{stab}_A&=\{h\in H : h\cdot A=A\}\\&=\{h\in H : hAh^{-1}=A\}\\&=H\cap \{g\in G : gAg^{-1}=A\}\\&=H\cap N_A\end{align}$