Image of the identity under a group homomorphism is the identity

For any group homomorphism \(f: G \to H\), we have \(f(e_G) = e_H\) where \(e_G\) is the identity of \(G\) and \(e_H\) the identity of \(H\).

Indeed, \(f(e_G) f(e_G) = f(e_G e_G) = f(e_G)\), so premultiplying by \(f(e_G)^{-1}\) we obtain \(f(e_G) = e_H\).