homomorphisms of simple groups
The kernel of a homomorphism must be a normal subgroup. Since is simple, there are only two possibilities: either the kernel is all of of it consists of the identity. In the former case, the homomorphism will map all elements of 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, 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|
|Date of creation||2013-03-22 15:41:59|
|Last modified on||2013-03-22 15:41:59|
|Last modified by||rspuzio (6075)|