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State/First isomorphism theorem for groups
Let $Im(\phi)$ denote the image of $\phi$, and let $Ker(\phi)$ denote the kernel of $\phi$. Then, the homomorphism $\phi$ descends to an isomorphism: $$\bar \phi \colon G / Ker(\phi) \rightarrow Im(\phi),$$ where $G/Ker(\phi)$ denotes the quotient group.
- If $\phi$ is surjective, then $G / Ker(\phi)$ is isomorphic to $H$.
- If $\phi$ is injective, then $G$ is isomorphic to $Im(\phi)$.
This statement logically relies on the following definitions and statements: Def/Group, Def/Group homomorphism, Def/Image, Def/Kernel (group), Def/Quotient group, Def/Inclusion function, State/Kernel criterion for injectivity of group homomorphisms
The following statements and definitions rely on the material of this page: State/Cyclic groups are classified by their order.
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