QUESTION : Let $G$ be a group, let $X$ be a set, and let $H$ be a subgroup of $G$. Let $$N = \bigcap_{g\in G} gHg^{-1}$$ Show that $N$ is a normal subgroup of $G$ cointained in $H$.
MY ATTEMPT: I began by asking myself precisely what $\bigcap_{g\in G} gHg^{-1}$ means. I concluded that it must mean that if $g_1, g_2, g_3, … , g_n \in G$ then $N$ might just be $$ g_1H{g_1}^{-1} \cap g_2H{g_2}^{-1} \cap g_3H{g_3}^{-1} \cap … g_nH{g_n}^{-1}$$ which I figured is actually just $H$ because that's the only element that all those elements have in common.
Therefore I figured that $N=H$.
Now my problem comes in showing that $H$ is a normal subgroup of $G$. I've never been good at that.
Best Answer
Hint: $H\leq G$ and let $\Omega$ be the set of all $Ha$ where $a\in G$. Define an action like: $$(Ha)^x=Hax,~~ Ha,Hax\in\Omega;~~x\in G$$ By this action we see that the stabilizer of $Ha$ for example is $a^{-1}Ha$. Now try to show that the map $x\mapsto\bar{x},~~\bar{x}(Ha)=Hax$ is a homomorphism with the kernel $N$.