If $\phi$ is a morphism between groups $G$ and $H$, is $G$ isomorphic to $$ker(\phi)\times im(\phi)$$ ? Why ? Thanks.
2026-04-05 16:17:54.1775405874
isomorphism between group and product of kernel by image
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No. Consider $\phi : \mathbb{Z}_4 \to \mathbb{Z}_2$, $a \mapsto 2a$. We have $\ker\phi = \operatorname{im}\phi = \mathbb{Z}_2$, but $\mathbb{Z}_4 \not\cong \mathbb{Z}_2\times\mathbb{Z}_2$.
What is true however is that $G/\ker\phi \cong \operatorname{im}\phi$. This is known as the First Isomorphism Theorem which holds for both groups and modules.