Let $G$ be an abelian group, and $H$ and $N$ is a subgroup. And assume $$ G / H \cong N \ . $$ Are these assumptions sufficient to show that $G$ is direct sum of H and N?
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2026-03-25 06:00:41.1774418441
abelian group, direct summand
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No. There is a counter example.
$H=N= \mathbb{Z} / 2 \mathbb{Z}$, $G = \mathbb{Z}/ 4 \mathbb{Z}$.