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)
Is isomorphism of two subgroups, one of them normal, enough to guarantee that the other is normal as well?
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abstract-algebra
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1My answer [to this question](http://math.stackexchange.com/q/63041/11619) contains counterexamples. As an exercise I invite you to find two subgroups of the dihedral group of 8 elements (= the symmetries of a square). Both are expected to be cyclic of order two. One central (hence normal) the other not. – 2011-09-11
2 Answers
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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.
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0@jspecter: I now see. Beautiful :-) Thanks a lot! – 2011-09-11
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There are counterexamples with two elements in the group $\mathbb Z/2\mathbb Z\times S_3$.