Group homomorphism (group theory)

A group homomorphism is a mapping from one group to another in a structure-preserving manner.

Definition

A group homomorphism from group $(G,{\cdot }_G,1_G)$ to group $(H,{\cdot }_H,1_H)$ is a function $\phi :G\to H$ subject to the following constraints:

• identity is preserved: $\phi 1_G=1_H$;

• products are preserved: for any $g,h\in G$, $\phi (g{\cdot }_{G}h)=\phi g{\cdot }_H\phi h$.