Group Homomorphism

up:: Group (Mathematics)

A function $\phi :G\to G^{\prime }$ between groups $(G,{\cdot }_G)$ and $(G^{\prime },{\cdot }_{G^{\prime }})$ is said to be a homomorphism if it preserves both groups’ operations.

That is,

$$\forall g,h\in G:\phi (g{\cdot }_{G}h)=\phi (g){\cdot }_{G^{\prime }}\phi (h)$$

Corollaries

1. Homomorphisms preserve neutral elements: $\phi (g)=\phi (g{e}_G)=\phi (g)\phi (e_G)⟹e_{G^{\prime }}=\phi (e_G)$
2. Homomorphisms preserve inverse operations: $\phi (g)\phi (g{)}^{-1}=e_{G^{\prime }}=\phi (g{g}^{-1})=\phi (g)\phi (g^{-1})⟹\phi (g^{-1})=\phi (g{)}^{-1}$

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References

• Group homomorphism - Wikipedia