homomorphisms of simple groups

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.

Title homomorphisms of simple groups HomomorphismsOfSimpleGroups 2013-03-22 15:41:59 2013-03-22 15:41:59 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Theorem msc 20E32