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Definition: Let G and G' be groups, written with multiplicative notation. A homomorphism f:G -> G' is a map from G to G' such that for all a and b in G, f(ab)=f(a)f(b). The left side of this means first multiply a and b in G, then send the product to G' using the map f, while the right side means first send and b individually to G' using the map f, then multiply their images in G'. Intuitively, a homomorphism is a map that is compatible with the laws of composition in the two groups, and it provides a way to relate different groups.
Source: Algebra, second edition by Michael Artin