In Appendix B2 of Fourier Analysis on Groups by Walter Rudin one can read:
A homomorphism of a group $G$ into a group $G_1$ is a map $\phi$ of $G$ into $G_1$ such that $$\phi\left(x+y\right)=\phi\left(x\right)+\phi\left(y\right)\qquad\left(x,y\in G\right).$$ A homomorphism which is one-to-one is an isomorphism.
Is it not standard to require $\phi$ to be surjective?