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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.

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    The error in your last sentence is that isomorphic groups have the same properties as groups themselves (being cyclic, being abelian, having 4 subgroups of order 6, etc.), not when viewed inside other groups. There is no reason isomorphic objects must appear the same relative to properties inside some common larger object, and being a normal subgroup is such a property. For instance, if h and k are elements of order 2 in G then = {1,h} and = {1,k} are subgroups of G. If h is in the center of G and k is not then is normal in G and is not.2011-10-19

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