Let $G$ be a group and $H$ be a normal subgroup of $G$. Assume that $H$ is finite and cyclic. Show that every subgroup of $H$. is also normal in $G$.$$
I am trying to prove this problem, but absolutely cannot make any progress. I know that $H$ is generated by an element $a\in G$ with the order of $H$ being the order of $a$.
I also know that any subgroup $K$ of $H$ will be cyclic as well. I cannot seem to make any further progress from this.. any help would be great.