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# State/First isomorphism theorem for groups

Proposition: (First isomorphism theorem for groups) Suppose that $G$ and $H$ are groups, and $\phi \colon G \rightarrow H$ is a group homomorphism.

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.

In particular,

• If $\phi$ is surjective, then $G / Ker(\phi)$ is isomorphic to $H$.
• If $\phi$ is injective, then $G$ is isomorphic to $Im(\phi)$.

## Logical Connections

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.

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Basic group theory