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The theorem says that if I have a group homomorphism, then the kernel is normal and the image is isomorphic to the domain group modulo the kernel.

Now, suppose I have $G/K \cong{H}$ where $G$ and $H$ are group and $K<G$. Can I say there exists and epimomorphism from $G$ to $H$ where $K$ is the kernel?

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The canonical projection $G \to G/K$ composed with your isomorphism does the job.

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  • $\begingroup$ Thank you, that was really helpful. $\endgroup$ – Meitar Abarbanel Jan 31 '15 at 8:30

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