Possible Duplicate:
Is isomorphism of two subgroups, one of them normal, enough to guarantee that the other is normal as well?
Suppose $G$ is a group, $H$ is a normal subgroup of $G$ and $K$ is a subgroup of $G$. Suppose also that $H$ and $K$ are isomorphic. Does it follow that $K$ is a normal subgroup of $G$? I've tried extending a isomorphism $\phi: H \rightarrow K$ to a surjective homomorphism $f: G \rightarrow G$ (where $f(h) = \phi(h)$ for all $h \in H$). Then $f(H) = K$ would be normal in $G$, but I didn't get anywhere with this approach. I also tried thinking of counterexamples, but I doubt one exists since intuitively thinking you would believe that isomorphic subgroups have the same properties.