TITLE: $G$ is a group $,H \cong K$, then is $G/H \cong G/ K$?
QUESTION [5 upvotes]: $G$ is a group with subgroups $H$ and $K$ such that $,H \cong K$, then is $G/H \cong G/ K$?

REPLY [18 votes]: 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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