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Its seems true but I did not see in any group theory book. Let $G\times_{c} H$ be product of two abelian groups (both are infinite). Then $I$ is subgroup of product iff $I = A\times_{c} B$ where $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$. Thanks

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up vote 9 down vote accepted

This is not true: One has the diagonal embedding $G \to G \times G$ by $g \mapsto (g, g)$.

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this is a nice example – Gastón Burrull Feb 24 '13 at 19:28

You can find very detailed information about subgroups of direct product of groups (not necessarily Abelian) in:

Bauer K., Sen D., Zvengrowski P. A Generalized Goursat Lemma. (2011)

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