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$G$ is a group with subgroups $H$ and $K$ such that $,H \cong K$, then is $G/H \cong G/ K$?

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    Answered by/ Possible Duplicate of http://math.stackexchange.com/questions/7720/finite-group-with-isomorphic-normal-subgroups-and-non-isomorphic-quotients2012-06-01

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No. Consider $G = (\mathbb{Z},+)$, $H= (2\mathbb{Z},+)$ and $K= (4\mathbb{Z},+)$. Note that $H$ and $K$ are isomorphic by the mapping $z \to 2z$.

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    @JonasMeyer: I am not here to *impress* anyone here with my typing speed :)2012-05-31