homomorphisms of simple groups

If a group $G$ is simple, and $H$ is an arbitrary group then anyhomomorphism of $G$ to $H$ must either map all elements of $G$ to theidentity of $H$ or be one-to-one.

The kernel of a homomorphism must be a normal subgroup. Since $G$ issimple, there are only two possibilities: either the kernel is all of$G$ of it consists of the identity. In the former case, thehomomorphism will map all elements of $G$ to the identity. In thelatter case, we note that a group homomorphism is injective iff the kernelis trivial.

This is important in the context of representation theory. In thatcase, $H$ is a linear group and this result may be restated as sayingthat representations of a simple group are either trivial or faithful.