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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.

| | | | | created: 2019-08-03 07:39:48 | modified: 2019-08-03 11:05:31 | by: bookofproofs | references: [677]