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Theorem: First Isomorphism Theorem for Groups

Let $(G,\ast)$, $(H,\cdot)$ be groups and let $f:G\to N$ be a group homomorphism with the kernel $\ker(f).$ Then $(G/\ker(f),\circ)$ is a factor group $(G/\ker(f),\circ)$ and the function $$\phi:G/\ker(f)\to \operatorname{im}(f),\quad \phi(a\ker(f))=f(a)$$ between this factor group and the group of the image $\operatorname{im}(f)$ is an isomorphism, i.e. both groups are isomorphic.

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References

Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013