TITLE: Is isomorphism of two subgroups, one of them normal, enough to guarantee that the other is normal as well?
QUESTION [6 upvotes]: Let $G$ be a group and $H$, $K$ subgroups such that $H$ is normal in $G$ and $H$, $K$ are isomorphic. After some thought i intuitively concluded that $K$ is not necessarilly normal in $G$. Is that the case? Any rigorous argument? (e.g. counterexample)

REPLY [6 votes]: Yes. Take any group $G'$ with a non-normal subgroup $K'$ and consider $G = G' \oplus K'.$ Then $H = 1 \oplus K'$ is normal in $G$ and isomorphic to the non-normal subgroup $K$ obtained from embedding $K'$ in the first coordinate.

REPLY [2 votes]: There are counterexamples with two elements in the group $\mathbb Z/2\mathbb Z\times S_3$.