I was doing a course on algebra and had this question written in my notes.
Prove that if $H$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$, then $K$ may not be a normal subgroup of $G$.
Now as I understand to prove a subgroup $K$ normal to $G$ I have to do $g^{-1}kg$ belongs to $K$. Clearly, This equation will be satisfied for all $g$ belonging to $H$ (as $K$ is a normal subgroup to H) but not necessarily for $g$ belonging to $G-H$.
Formally, I am at a loss how to show this that there may exist an element which will not satisfy this. Moreover I feel I am missing something as I have still not used $H$ is a subgroup.