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Whilst I understand the classification of the finitely generated abelian groups, this had me wondering whether there is a subgroup $H$ of a general (necessarily infinitely generated) abelian group $G$ such that $H$ is not isomorphic to any quotient $G/N$ of $G$.

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    $\begingroup$ To complete Eric's answer; exercise: for an abelian group $H$, there exists an overgroup $G$ such that $H$ is not isomorphic to a quotient of $G$ if and only if $H$ is non-divisible. $\endgroup$ – YCor Apr 2 '18 at 22:29
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For a very simple example, let $G=\mathbb{Q}$ and $H=\mathbb{Z}$. Since $\mathbb{Q}$ is divisible (for any $x\in\mathbb{Q}$ and any nonzero integer $n$, there exists $y\in\mathbb{Q}$ such that $x=ny$), any quotient of $\mathbb{Q}$ is also divisible, so $\mathbb{Z}$ is not isomorphic to any quotient of $\mathbb{Q}$.

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  • $\begingroup$ D'oh, yep, thank you! $\endgroup$ – Kelechi Nze Apr 2 '18 at 20:01

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