Group Homomorphism

A function $f:G\to H$ between two groups $(G,{\cdot }_G,e_G,{\_}^{-1_G})$ and $(H,{\cdot }_H,e_H,{\_}^{-1_H})$ is a group homomorphism if it preserves the group operation: $\forall x,y\in G,f(x{\cdot }_{G}y)=f(x){\cdot }_{H}f(y)$ It follows from this property that $f(e_G)=e_H$ and $f(x^{-1_G})=f(x{)}^{-1_H}$.