homomorphisms of simple groups

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

The kernel of a homomorphism must be a normal subgroupMathworldPlanetmath. 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 groupMathworldPlanetmathPlanetmath are either trivial or faithful.

Title homomorphisms of simple groups
Canonical name HomomorphismsOfSimpleGroups
Date of creation 2013-03-22 15:41:59
Last modified on 2013-03-22 15:41:59
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 4
Author rspuzio (6075)
Entry type Theorem
Classification msc 20E32