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# Def/Group homomorphism

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Definition of Group homomorphism: Suppose that $G$ and $H$ are groups. A group homomorphism (usually called a homomorphism, when the context is clear) from $G$ to $H$ is a function $f \colon G \rightarrow H$, such that, for all $g_1, g_2 \in G$, $$f(g_1 g_2) = f(g_1) f(g_2).$$

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Group, State/Cancellation in groups

The following statements and definitions logically rely on the material of this page: Def/Kernel (group), State/First isomorphism theorem for groups, State/Homomorphic images of subgroups are subgroups, and State/Kernel criterion for injectivity of group homomorphisms

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic group theory