[Math] A group homomorphism from a simple group is injective

Question

Let $G_1$ be a simple group, that is the only normal subgroups of $G_1$ are itself and the trivial subgroup. If $\phi : G_1 \rightarrow G_2$ is a group homomorphism, does that mean $\phi$ is injective? Could someone explain?

Answer 1

Looking at $\ker \phi\lhd G_1$ we note that $\phi$ is either injective or trivial.