Let $G$ be a group and $H$ be its normal subgroup. Is $G \cong H \times G/H$ ? Any counter-example if it is not true?
2026-03-26 07:45:15.1774511115
Isomorphism Between Group and Direct product of normal subgroup and Quotient Group
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No, actually this is quite rare situation, even among abelian groups. The simpliest counterexample is the cyclic group of order $4$: $G=\mathbb{Z}_4$, $H=\{0,2\}$. Note that $G/H\simeq\mathbb{Z}_2\simeq H$ but $G$ is not isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$ since the latter has no element of order $4$.