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homomorphisms of simple groups

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

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

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