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.

Answer 2

First of all let's make an observation.

Oss: Let $f:G \rightarrow H$ an homomorphism of groups. Then $\ker f$ is a normal subgroup of $G$.

In fact if $k \in \ker f$ for all $g \in G$ we have that $f(gkg^{-1})=f(g)f(k)f(g^{-1})=f(g)f(g^{-1})=e$. Then $g \ker f \, g^{-1} \subset \ker f $ that implies $\ker f$ is normal in $G$

Now in your case $G$ is simple so $\ker f$ can only be $\{e\}$ or all $G$, so the homomorphism is injective or is the null one.