Why is a particular element in a normal subgroup of G?

Question

This is a solution to an excercise:

No matter how I think about it I cannot see how $hkh^{-1}k^{-1}$ is an element of both $H$ and $K$

Answer 1

Since $K$ is a subgroup of $G$, $k \in G$. As $H$ is a group, $h^{-1} \in H$, and because it is a normal subgroup of $G$, so does $kh^{-1}k^{-1}$. Since $H$ is a group, $h(kh^{-1}k^{-1}) \in H$. Similar reasoning can be used to show that $hkh^{-1}k^{-1} \in K$. The solution provided is incorrect in the sense that the second expression shows that the element belongs to $H$, not $K$ (at least for the reasoning I have used in my explanation).

Answer 2

It looks to me like there's a typo in the first vs. second expression part. Since $K$ is normal, $hkh^{-1}$ is still in $K,$ and thus so is $hkh^{-1}k^{-1}.$ The second way of writing it (and the same argument) shows it is in $H.$