I have got the following task: Prove that if $H$ is a subgroup of the group $G$, then $K:=\bigcap_{g \in G} g H g^{-1}$ is a normal subgroup in $G$, it lies inside $H$ and contains each normal subgroup of $G$ with lies in $H$.
For the first task, I have already found an answer here. However, I am still unsure about the two remaining ones.
If I want to prove that $K$ lies in $H$, is it enough to use that since $H$ is a subgroup of $G$, multiplying any element of $H$ with an arbitrary $g \in G$ will always remain in $H$?
For the last one, I should maybe use the fact that the intersection of normal subgroups is also a normal subgroup?
Any help appreciated.
I already answered the first question in the comments above. Here is an answer to the second question:
Let $N\subseteq H$ be a normal subgroup of $G$. We want to prove $N\subseteq K$.
We have for any $g\in G$ that $gNg^{-1}\subseteq gHg^{-1}$. But $gNg^{-1}=N$, since $N$ is normal. Therefore $N\subseteq gHg^{-1}$.
Since $N$ is contained in each of the $gHg^{-1}$, it must be contained in their intersection, which is $K$.