Let \((G,\ast)\) and \((H,\cdot)\) be two groups with the respective identities \(e_G\) and \(e_H\) and \(f:G\rightarrow H\) be a group homomorphism. Then it has the following properties:
\[\begin{array}{cl} (1)&f(e_G)=e_H,\\ (2)&f(x^{-1})=f(x)^{-1}~~\forall x\in G. \end{array}\]
Proofs: 1
Examples: 1 2
Proofs: 3 4 5 6 7